The dramatists chose to portray only the beginning of the story of Caesar and Cleopatra in their play; they concentrate on the events of 48 BC. The play is set in Egypt; at its start, the Pharaoh Ptolemy XIII has sequestered his sister/wife/queen Cleopatra and has assumed sole rule of the kingdom, and the Battle of Pharsalia has not yet occurred. By the play's end, Caesar has deposed Ptolemy and placed Cleopatra in sole possession of the Egyptian crown. The play's Prologue specifically states that the work shows a virginal "Young Cleopatra...and her great Mind / Express'd to the height...." Some of the famous aspects of the story are reproduced in the play: Cleopatra has herself delivered to Caesar in Act III, though enclosed in a "packet" rather than rolled up in a rug.
The playwrights chose to concentrate much of their attention on the figure of Lucius Septimius, the Roman officer who betrayed, murdered, and decapitated Pompey the Great when Pompey landed in Egypt after his Pharsalia defeat (events depicted in Act II). Septimius is the "false one" of the title, and his prominence comes close to turning the work into a "villain play." Yet Septimius is portrayed as lacking any redeeming or sympathetic quality, making him a weak prop on which to mount a drama. The authors' choice in this matter may have been dictated by their desire to comment on contemporaneous political events; in this interpretation, the Pompey of the play represents Sir Walter Raleigh, executed in 1618, while the loathsome reprobate Septimius stands for Raleigh's primary accuser, Sir Lewis Stukeley.
Critics have seen the influence of Shakespeare's Antony and Cleopatra in The False One, and have suggested that the portrayal of Septimius was partially modelled on Shakespeare's Enobarbus. The False One is heavily dominated by political material, rather than dramatic realisations of its characters; for some critics, the split in the play's focus among Cleopatra, Caesar, and Septimius prevents the play from cohering into an effective dramatic whole.

Question: Who rules Egypt during this time?
Answer: Pharaoh Ptolemy XIII rules Egypt. 
Alternative answer: Ptolemy XIII