Documentation, or the lack there of has always been a pet peeve of mine. I have inherited many of these.

First step is to resist the temptation to strangle the person before you. I assume they are leaving the company as opposed to being promoted for not documenting their work.

Second step is to have great contact information about the previous owner. Ideally develop a list of all people who touched the project. This will most likely be a list of two people, you and your predecessor.

Minimal things you need to know:
1) Where are the files?
2) How do you build the project?
3) What is the typical input?
4) What is the typical output?
5) Are there any legal issue?

IMHO most projects have a collection of major parts. Start documenting those parts. Personally, I like MS One Note, but a pad of paper is better than nothing. After each major part is documented – as in a paragraph on each, start delving into each file. Create a paragraph about each.

Next I would try and document the requirements. It is critical to know what it is supposed to do. This should cover details of the goes into and goes out of. Also essential is legal issues. Everything legal you must inform management of, and have a documented reply! If there are no legal issues, you still need a confirmation email saying that. Do not put your freedom at risk, because ‘of course there are no legal issue what do you think we are building’. You should not become the patsy for someone else’s mistakes.

Bring all of this to your boss. Your boss should have some idea of the ‘Hell’ you have been inflicted with. Make sure your boss realizes: the ‘un professional’ people before you could not be bothered to document the bare minimal, but now you, ‘the professional’ has done so. Also tell them how many man hours this took.

If it looks like this is going to take more than a man week, enlist help from others to get the above done. Get your boss to agree to ‘Compensatory time off’ or ‘over time’ to get this hand over done. Since you only have a month before the person leaves. Stress to your boss, an extra man day spend this month will save a man week in the future.

Next understand how to make changes. Make some kind of needed change, and walk through the implementation. I recommend for the first one something like change of wording somewhere.

If you get help, have people document routines. 1) Purpose of routine 2) inputs 3) outputs 4) side effects.

Next document known problems.

Also obtain a list of customers who use product.

Are there test scripts? How do you run them? How do you change them.

When the month is over, analyze what you have documented.
1) Is there enough that you will not be spending most of your time saying WTF?
2) Would it be easier to re write the code? Try to resist this, as it might be easier in your mind it is often harder, and not rewriting is part of this exercise for the last month.
3) Does your boss appreciate what you have done, and how hard this is? Given the state of the project I would assume not. If your boss had any skills it would not have been left that way. Is there a bonus in your future for this?
I would also consider looking for a new job. Undocumented development is a key sign for a poorly run organization.