<PAGE>

                                                                   EXHIBIT 10.34

                                                            [EXECUTION COPY]

                               BAIN/ACR, L.L.C.
                       EXECUTIVE UNIT PURCHASE AGREEMENT
                       ---------------------------------

          THIS EXECUTIVE UNIT PURCHASE AGREEMENT (this "Agreement") is made as
                                                        ---------
of May 21, 1999, by and between Bain/ACR, L.L.C., a Delaware limited liability
company ("Bain/ACR"), Anthony Crane Rental Holdings, L.P., a Pennsylvania
          --------
limited partnership ("Holdings"), ACR Management, L.L.C., a Delaware limited
                      --------
liability company and the general partner of Holdings and Anthony Crane ("ACR
                                                                          ---
Management") and Joseph Connelly ("Executive").  Any capitalized terms used
----------                         ---------
herein and not otherwise defined shall have the meanings assigned to them in
Section 5 hereof.

          WHEREAS, Bain/ACR, Holdings, ACR Management and Executive desire to
enter into this Agreement (i) to provide for the sale to Executive by Bain/ACR
of 9,928.35 of Holding's Class A Common Units (the "Class A Units"), 1,103.15 of
                                                    -------------
Holding's Class L Common Units (the "Class L Units") a 0.25% Membership Interest
                                     -------------
in ACR Management (the "Membership Interest", and collectively with any Class A
                        -------------------
Units and Class L Units, the "Purchased Units"), (ii) to provide for the
                              ---------------
issuance to Executive by Holdings of 11,142.93 of Holding's Class B Common Units
(the "Class B Units") and 1,103.15 of Holding's Class C Common Units (the "Class
      -------------                                                        -----
C Units", and collectively with such Class B Units, the "Incentive Units" ) and
-------                                                  ---------------
(iii) to provide for certain rights and obligations of the parties with respect
to the Purchased Units and the Incentive Units (collectively, the "Executive
                                                                   ---------
Units").
-----

          NOW THEREFORE, in consideration for the promises contained herein and
the mutual obligations of the parties hereto, the receipt and sufficiency of
which are hereby acknowledged, Bain/ACR, Holdings, ACR Management and the
Executive hereto agree as follows:

     1.  Purchase and Sale of Executive Units.
         ------------------------------------

          (a) Upon execution of this Agreement, (i) Executive shall purchase,
and Bain/ACR  shall sell, 9,928.35 Class A Units at a price of $1.00 per unit
and 1,103.15 Class L Units at a price of $81.00 per unit, (ii) Executive shall
purchase, and Bain/ACR shall sell, a 0.25% Membership Interest in ACR Management
for $3,216.46, and (iii) and Executive shall purchase, and Holdings shall sell,
11,142.93 Class B Units at a price of $0.70 per unit and 11,142.93 Class C Units
at a price of $0.40 per unit.  Holdings shall deliver to Executive an executed
copy of the Fourth Amended and Restated Limited Partnership Agreement of
Holdings (the "Partnership Agreement") indicating Executive's ownership of such
               ---------------------
Class A Units, Class L Units, Class B Units and Class C Units and an executed
copy of the First Amended and Restated Limited Liability Company Agreement of
ACR Management (the "LLC Agreement") indicating Executive's ownership of such
                     -------------
Membership Interest, and Executive shall deliver a cashier's check or wire
transfer of funds in the aggregate amount of $102,500 to Bain/ACR and a
cashier's check or wire transfer of funds in the aggregate amount of $12,256.57
to Holdings.

          (b) Within 30 days after Executive purchases any Executive Units from
Bain/ACR, Executive shall make an effective election with the Internal Revenue
Service under

                                      -1-
<PAGE>

Section 83(b) of the Internal Revenue Code and the regulations promulgated
thereunder in the form of Exhibit A attached hereto.
                          ---------

          (c) Concurrently with the execution of this Agreement, Executive will
enter into an Employment Agreement with Anthony Crane in the form of Exhibit B
                                                                     ---------
attached hereto (the "Employment Agreement").
                      --------------------

     2.  Representations and Warranties; Acknowledgments.
         -----------------------------------------------

          (a) Representations and Warranties by Executive.  In connection with
              -------------------------------------------
the purchase and sale of the Executive Units hereunder, Executive represents and
warrants to Bain/ACR, Holdings and ACR Management that:

               (i) The Executive Units to be acquired by Executive pursuant to
     this Agreement shall be acquired for Executive's own account and not with a
     view to, or intention of, distribution thereof in violation of the
     Securities Act, or any applicable state securities laws, and the Executive
     Units shall not be disposed of in contravention of the Securities Act or
     any applicable state securities laws.

               (ii) Executive is an executive officer of Anthony Crane, is
     sophisticated in financial matters and is able to evaluate the risks and
     benefits of the investment in the Executive Units.

               (iii)  Executive is able to bear the economic risk of his
     investment in the Executive Units for an indefinite period of time because
     the Executive Units have not been registered under the Securities Act and,
     therefore, cannot be sold unless subsequently registered under the
     Securities Act or an exemption from such registration is available.

               (iv) Executive has had an opportunity to ask questions and
     receive answers concerning the terms and conditions of the offering of
     Executive Units and has had full access to such other information
     concerning Holdings and ACR Management as he has requested.  Executive has
     also reviewed, or has had an opportunity to review, the following
     documents: (A) the Partnership Agreement; (B) the LLC Agreement; (C) the
     loan agreements, notes and related documents with the senior and
     subordinated lenders of Holdings; and (D) Holdings' audited and unaudited
     financial statements.

               (v) The execution, delivery and performance of this Agreement by
     Executive do not and shall not conflict with, breach, violate or cause a
     default under any contract, agreement, instrument, order, judgment or
     decree to which Executive is a party or by which he is bound and upon the
     execution and delivery of this Agreement by Bain/ACR, Holdings and ACR
     Management, this Agreement shall be the legal, valid and binding obligation
     of Executive, enforceable in accordance with its terms.

               (vi) Executive is not a party to or bound by any employment
     agreement, noncompete agreement or confidentiality agreement with any
     person or entity other than Anthony Crane.

                                      -2-
<PAGE>

               (vii)  Executive has consulted with independent legal counsel
     regarding his rights and obligations under this Agreement and he fully
     understands the terms and conditions contained herein.

          (b)  Acknowledgments.
               ---------------

               (i) As an inducement to Bain/ACR and Holdings to sell the
     Executive Units to Executive, as a condition thereto, Executive
     acknowledges and agrees that:

                    (A) neither the sale of the Executive Units to Executive nor
          any provision contained herein shall entitle Executive to remain in
          the employment of Anthony Crane or affect the right of Anthony Crane
          to terminate Executive's employment at any time; and

                    (B) neither Bain/ACR nor Holdings shall have any duty or
          obligation to disclose to Executive, and Executive shall have no right
          to be advised of, any material information regarding Holdings and its
          Subsidiaries at any time prior to, upon or in connection with the
          repurchase of Executive Units upon the termination of Executive's
          employment with Anthony Crane or as otherwise provided hereunder.

               (ii) Bain/ACR, Holdings and Executive acknowledge and agree that
     this Agreement has been executed and delivered, and the Executive Units
     have been sold hereunder, in connection with and as a part of the
     compensation and incentive arrangements between Holdings and Executive.

     3.  Right to Purchase Executive Units Upon Termination of Employment.
         ----------------------------------------------------------------

          (a) Repurchase Option.  In the event that Executive is no longer
              -----------------
employed by Anthony Crane for any reason (the date of such termination being
referred to herein as the "Termination Date"), the Executive Units, whether held
                           ----------------
by Executive or one or more Permitted Transferees, will be subject to repurchase
by Holdings, Bain/ACR and/or ACR Management pursuant to the terms and conditions
set forth in this Section 3 (the "Repurchase Option").
                                  -----------------

          (b) Termination Other than for Cause.  If Executive is no longer
              --------------------------------
employed by Anthony Crane as a result of Executive's death or permanent
disability (as determined by the Board in its good faith judgment), Executive's
termination by Anthony Crane without Cause or voluntary termination, then on or
after the Termination Date, (1) Holdings may elect to purchase all or any
portion of (i) the Purchased Units (other than the Membership Interest) and the
Vested Incentive Units at a price per unit equal to the Fair Market Value
thereof as determined as of a date set by the Board within thirty (30) days
prior to the delivery of the Repurchase Notice (as defined in subparagraph 3(d)
below) and (ii) the Unvested Incentive Units at a price per unit equal to the
lower of their Original Cost or the Fair Market Value thereof determined as
described in clause 3(b)(1) above and (iii) ACR Management may elect to purchase
all or any portion of the Membership Interest at a price equal to the Fair
Market Value thereof as determined as of a date set by the Board within thirty
(30) days prior to the delivery of the Repurchase Notice.

                                      -3-
<PAGE>

          (c) Termination for Cause.  If Executive is no longer employed by
              ---------------------
Anthony Crane as a result of Executive's termination for Cause, then on or after
the Termination Date, Holdings may elect to purchase all or any portion of the
Executive Units (other than the Membership Interest) and ACR Management may
elect to purchase the Membership Interest at a price equal to the lower of the
Original Cost thereof or the Fair Market Value thereof determined as described
in clause 3(b)(1) above.

          (d) Repurchase Procedures.  Holdings or ACR Management may elect to
              ---------------------
exercise the right to purchase all or any portion of the Executive Units
pursuant to the Repurchase Option by delivering written notice (the "Repurchase
                                                                     ----------
Notice") to the holder or holders of Executive Units within 180 days after
------
Executive's Termination Date.  The Repurchase Notice will set forth the number
of Executive Units to be acquired from such holder(s), the aggregate
consideration to be paid for such units and the time and place for the closing
of the transaction.  Holdings may elect to purchase all or any portion of the
Unvested Incentive Units without or before purchasing any Vested Incentive
Units.  If any of the Executive Units are held by Permitted Transferees of
Executive, Holdings or ACR Management shall purchase the units or membership
interest elected to be purchased from such holder(s) of Executive Units pro rata
according to the number of Executive Units held by such holder(s) at the time of
delivery of such Repurchase Notice (determined as nearly as practicable to the
nearest unit).  If both Unvested Units and Vested Units are to be purchased by
Holdings or General Partner and Executive Units are held by Permitted
Transferees of Executive, the number of Unvested Units and Vested Units to be
purchased will be allocated among such holders pro rata according to the total
number of Executive Units to be purchased from such person.

          (e)  Investors' Rights.
               -----------------

               (i) If for any reason Holdings and/or ACR Management does not
     elect to purchase all of the Executive Units pursuant to the Repurchase
     Option prior to the 180th day following the Termination Date, Bain/ACR will
     be entitled to exercise the Repurchase Option, in the manner set forth in
     this Section 3, for the Executive Units Holdings and/or ACR Management has
     not elected to purchase (the "Available Units").  As soon as practicable,
                                   ---------------
     but in any event within thirty (30) days after Holdings determines that
     there will be any Available Units, Holdings or ACR Management will deliver
     written notice (the "Option Notice") to Bain/ACR setting forth the number
                          -------------
     of Available Units and the price for each Available Unit.

               (ii) Bain/ACR will initially be permitted to purchase the
     Available Units by delivering written notice to Holdings and ACR Management
     within twenty (20) days after receipt of the Option Notice from Holdings
     and/or ACR Management (such 20-day period being referred to herein as the

     "Investor Election Period").
     -------------------------

               (iii)  As soon as practicable but in any event within five (5)
     business days after the expiration of the Investor Election Period,
     Holdings or ACR Management will, if necessary, notify the holder(s) of
     Executive Units as to the number of Executive Units being purchased from
     the holder(s) by Bain/ACR (the "Supplemental Repurchase Notice").  The
                                     ------------------------------
     Supplemental Repurchase Notice will set forth the number of Executive Units
     Holdings,

                                      -4-
<PAGE>

     General Partner and Bain/ACR will acquire from such holder(s), the
     aggregate consideration to be paid for such units and the time and place of
     the closing of the transaction.

          (f) Closing.  The closing of the transactions contemplated by this
              -------
Section 3 will take place on the date designated by Holdings or ACR Management
in the Repurchase Notice or the Supplemental Repurchase Notice, as the case may
be, which date will not be more than ninety (90) days after the delivery of such
notice.  Holdings, ACR Management and/or Bain/ACR, as the case may be, will pay
for the Executive Units to be purchased pursuant to the Repurchase Option by
delivery of, in the case of  Bain/ACR, a check payable to the holder of
Executive Units, and in the case of Holdings or ACR Management (i) a check
payable to the holder of such Executive Units, (ii) a note or notes payable in
three equal annual installments beginning on the first anniversary of the
Termination Date and bearing interest (payable quarterly) at a rate per annum
equal to 8% or (iii) both (i) and (ii) in the aggregate amount of the purchase
price for such units.  Any notes issued by Holdings pursuant to this paragraph
3(f) shall be subject to any restrictive covenants to which Holdings is subject
at the time of such purchase.   Holdings, ACR Management and/or Bain/ACR, as the
case may be, will receive customary representations and warranties from each
seller regarding the sale of the Executive Units, including, but not limited to,
the representation that such seller has good and marketable title to the
Executive Units to be transferred free and clear of all liens, claims and other
encumbrances, other than liens pursuant to the Pledge Agreement.

          (g) Termination of Repurchase Right. The rights of Holdings, ACR
              -------------------------------
Management and Bain/ACR to repurchase Executive Units pursuant to this Section 3
shall terminate upon the earlier of (i) a Sale of the Company or (ii) a Public
Offering.

          4.  Restrictions on Transfer. The parties hereby agree that the
              ------------------------
Executive Units will be subject to the restrictions on Transfer and other
provisions contained in the Securityholders Agreement and will be considered
"Other Securities" for purposes of the Securityholders Agreement.

          5.  Definitions.
              -----------

          "Anthony Crane" means Anthony Crane Rental, L.P., a Pennsylvania
           -------------
limited partnership and a wholly-owned subsidiary of Holdings.

          "Affiliate" shall have the meaning assigned to it in the Partnership
           ---------
Agreement.

          "Board" shall mean the Board of Managers of ACR Management.
           -----

          "Cause" shall have the meaning assigned to it in the Employment
           -----
Agreement.

          "Common Units" is defined in the Partnership Agreement and includes
           ------------
any equity securities issued or issuable directly or indirectly with respect to
such Common Units by way of any dividend or split or exchange or in connection
with a combination of units, recapitalization, merger, consolidation or other
reorganization.

          "Executive Units"  means collectively the Class A Units, Class L
           ---------------
Units, the Class B Units, the Class C Units and the Membership Interest acquired
by the Executive pursuant to Section 1.  Such Units shall continue to be
Executive Units in the hands of any holder other than Executive

                                      -5-
<PAGE>

(except for Holdings, Bain/ACR and transferees in a Public Sale), and except as
otherwise provided herein, each such other holder of Executive Units shall
succeed to all rights and obligations attributable to Executive as a holder of
Executive Units hereunder.  Executive Units shall include both vested and
unvested Executive Units and shall include interests in Holdings or ACR
Management issued with respect to Executive Units by way of any split, dividend
or recapitalization.

          "Fair Market Value" of each Executive Unit means the amount which each
           -----------------
such Unit would receive upon a complete liquidation of Holdings and ACR
Management following a sale of Holdings and ACR Management at their aggregate
market value as determined in good faith by the Board and Executive. If the
Board and Executive are unable to agree upon such market value, it shall be
determined in good faith by a nationally recognized investment banking
institution selected by the Board, the fees and expenses of which will be split
evenly between Holdings and ACR Management, on the one hand, and Executive, on
the other hand.

          "LLC Agreement" has the meaning assigned to it in Section 1(a) hereof.
           -------------

          "Original Cost" means $81.00 for each Class L Unit, $1.00 for each
           -------------
Class A Unit,  $0.70 for each Class B Unit, $0.40 for each Class C Unit and
$3,216.46 for the Membership Interest.

          "Partnership Agreement" has the meaning assigned to it in Section 1(a)
           ---------------------
hereof.

          "Permitted Transferee" has the meaning assigned to it in the
           --------------------
Securityholders Agreement.

          "Person" shall mean an individual, a partnership, a corporation, a
           ------
limited liability company, an association, a joint stock company, a trust, a
joint venture, an unincorporated organization and a governmental entity or any
department, agency, or political subdivision thereof.

          "Public Offering" means an offering of Holding's (or a corporate
           ---------------
successor's) equity securities to the public pursuant to an effective
registration statement under the Securities Act.

          "Public Sale" means any sale pursuant to a registered public offering
           -----------
under the Securities Act or any sale to the public pursuant to Rule 144
promulgated under the Securities Act effected through a broker, dealer or market
maker.

          "Recapitalization Agreement" means that certain Recapitalization
           --------------------------
Agreement, dated as of June 1, 1998 and amended as of July 22, 1998, by and
among Holdings, the purchasers listed on the Schedule of Purchasers and the
current owners listed on the Schedule of Current Owners attached thereto.

          "Sale of the Company" means (i) any sale of all or substantially all
           -------------------
(as defined in the Model Business Corporation Act) of the assets of Holdings and
its Subsidiaries on a consolidated basis in one transaction or series of related
transactions, (ii) any sale of all or substantially all of the Common Units in
one transaction or series of related transactions, excluding any sales of Common
Units in a Public Sale or (iii) a merger or consolidation which accomplishes one
of the foregoing.

                                      -6-
<PAGE>

          "Securities Act" means the Securities Act of 1933, as amended from
           --------------
time to time.

          "Securityholders Agreement" means that certain Securityholders
           -------------------------
Agreement, dated as of July 22, 1998 and as amended from time to time, among
Holdings and the signatories parties thereto.

          "Subsidiary" shall have the meaning assigned to it in the Partnership
           ----------
Agreement.

          "Transfer"  shall have the meaning assigned to it in the
           --------
Securityholders Agreement.

          "Units" means collectively the Class L Units, the Class A Units, the
           -----
Class B Units, the Class C Units and the Membership Interests.

          "Unvested Units" means any Executive Units which are not Vested Units.
           --------------

          "Unvested Incentive Units" means any Incentive Units which have not
           ------------------------
become Vested Incentive Units.

          "Vested Units" means any Purchased Units and any Vested Incentive
           ------------
Units.

          "Vested Incentive Units" means any Incentive Units which have become
           ----------------------
vested on a monthly basis in accordance with the following schedule, if as of
each such date Executive is employed by Anthony Crane:

                                      Cumulative Percentage
         Date                       of Incentive Units Vested
         ----                       -------------------------
Date of this Agreement                          0%
    August 1, 1999                             20%
    August 1, 2000                             40%
    August 1, 2001                             60%
    August 1, 2002                             80%
    August 1, 2003                            100%

Notwithstanding the foregoing, upon the occurrence of a Sale of the Company, so
long as Executive is employed by Anthony Crane as of the date on which such Sale
of the Company occurs, all Incentive Units which have not yet become vested
shall immediately become vested; provided that a merger or consolidation
                                 -------- ----
involving Holdings in which the holders of voting securities of Holdings own
more than 25% of the voting securities of the combined entity following such
transaction shall not be considered a Sale of the Company for purposes of the
immediately preceding clause.

          6.  Notices.  Any notice provided for in this Agreement must be in
              -------
writing and must be either personally delivered, mailed by first class mail
(postage prepaid and return receipt requested) or sent by reputable overnight
courier service (charges prepaid) to the Investors at the addresses indicated in
the Company's records and to the other recipients at the address indicated
below:

                                      -7-
<PAGE>

          Notices to Executive:
          --------------------

          Joseph Connelly
          400 Seco Road
          Monroeville, PA  15146

          Notices to Bain/ACR:
          -------------------

          Bain/ACR, L.L.C.
          c/o Bain Capital, Inc.
          Two Copley Place
          Boston, Massachusetts 02116
          Attn:  Robert Gay
                 Paul Edgerley
                 Andrew Balson

          Notices to Holdings:
          -------------------

          Anthony Crane Rental Holdings, L.P.
          400 Seco Road
          Monroeville, PA 15146
          Attn:  David Mahokey
                 Arthur J. Innamorato

          with a copy to:
          --------------

          Kirkland & Ellis
          200 East Randolph Drive
          Chicago, Illinois 60601
          Attn:  James L. Learner, P.C.

or such other address or to the attention of such other person as the recipient
party shall have specified by prior written notice to the sending party.  Any
notice under this Agreement shall be deemed to have been given when so delivered
or sent or, if mailed, five days after deposit in the U.S. mail.

          7.  General Provisions.
              ------------------

          (a) Transfers in Violation of Agreement.  Any Transfer or attempted
              -----------------------------------
Transfer of any Executive Units in violation of any provision of this Agreement
shall be void, and Holdings shall not record such Transfer on its books or treat
any purported transferee of such Executive Units as the owner of such Executive
Units for any purpose.

          (b) Severability.  Whenever possible, each provision of this Agreement
              ------------
shall be interpreted in such manner as to be effective and valid under
applicable law, but if any provision of this Agreement is held to be invalid,
illegal or unenforceable in any respect under any applicable law

                                      -8-
<PAGE>

or rule in any jurisdiction, such invalidity, illegality or unenforceability
shall not affect any other provision or any other jurisdiction, but this
Agreement shall be reformed, construed and enforced in such jurisdiction as if
such invalid, illegal or unenforceable provision had never been contained
herein.

          (c) Complete Agreement.  This Agreement, those documents expressly
              ------------------
referred to herein and other documents of even date herewith embody the complete
agreement and understanding among the parties and supersede and preempt any
prior understandings, agreements or representations by or among the parties,
written or oral, which may have related to the subject matter hereof in any way.

          (d) Counterparts.  This Agreement may be executed in separate
              ------------
counterparts, each of which is deemed to be an original and all of which taken
together constitute one and the same agreement.

          (e) Successors and Assigns.  Except as otherwise provided herein, this
              ----------------------
Agreement shall bind and inure to the benefit of and be enforceable by
Executive, Holdings, ACR Management, Bain/ACR and their respective successors
and assigns (including subsequent holders of Executive Units); provided that the
                                                               -------- ----
rights and obligations of Executive under this Agreement shall not be assignable
except in connection with a Permitted Transfer of Executive Units hereunder.

          (f) Arbitration.  Any controversy, dispute or claim arising out of or
              -----------
relating in any way to this Agreement that cannot be resolved by negotiations
between Bain/ACR, Holdings and Executive shall be settled by arbitration in
accordance with the terms and provisions of Section 8.02 of the Recapitalization
Agreement.

          (g) Choice of Law.  The partnership law of the Commonwealth of
              -------------
Pennsylvania shall govern all questions concerning the relative rights of
Holdings and its partners.  The limited liability company law of the State of
Delaware shall govern all questions concerning the relative rights of ACR
Management and its members.  All issues and questions concerning the
construction, validity, enforcement and interpretation of this Agreement and the
exhibits and schedules hereto shall be governed by, and construed in accordance
with, the laws of the Commonwealth of Pennsylvania, without giving effect to any
choice of law or conflict of law rules or provisions (whether of the
Commonwealth of Pennsylvania or any other jurisdiction) that would cause the
application of the laws of any jurisdiction other than the Commonwealth of
Pennsylvania.

          (h) Remedies.  Each of the parties to this Agreement shall be entitled
              --------
to enforce its rights under this Agreement specifically, to recover damages and
costs (including reasonable attorney's fees) caused by any breach of any
provision of this Agreement and to exercise all other rights existing in its
favor.  The parties hereto agree and acknowledge that money damages would not be
an adequate remedy for any breach of the provisions of this Agreement and that
any party may in its sole discretion apply to any court of law or equity of
competent jurisdiction (without posting any bond or deposit) for specific
performance and/or other injunctive relief in order to enforce or prevent any
violations of the provisions of this Agreement.

                                      -9-
<PAGE>

          (i) Amendment and Waiver.  The provisions of this Agreement may be
              --------------------
amended and waived only with the prior written consent of Bain/ACR, Holdings,
ACR Management and Executive.  The provisions of Section 3 may be amended and
waived only with the prior written consent of Bain/ACR, Holdings and ACR
Management.



                            *      *      *      *

                                      -10-
<PAGE>

          IN WITNESS WHEREOF, the parties hereto have executed this Executive
Unit Purchase Agreement on the date first written above.

                              BAIN/ACR, L.L.C.

                              By: ______________________________________

                              Its:  ______________________________________


                              ANTHONY CRANE RENTAL HOLDINGS, L.P.

                              By:   ACR Management, L.L.C.
                              Its:  General Partner

                              By: ______________________________________

                              Its:  ______________________________________


                              ACR MANAGEMENT, L.L.C.

                              By: ______________________________________

                              Its:  ______________________________________


                              _________________________________________
                              Joseph Connelly