In Antony and Cleopatra, why does Egypt dominate Act 1?
I'm not sure that Egypt dominates Act I, unless by Egypt, you mean Cleopatra, since she is Pharaoh and, as such, would have been seen as "Egypt" incarnate. However, it is certainly worth examining why Shakespeare sets the opening scenes in Egypt when so much of the talk is of what is going on in Rome.
First, Shakespeare often chose exotic or foreign or supernatural locations for his plays. His audiences (in the same way we flock to movies of vampires or science fiction or futuristic settings) would have been lured to the playhouse by the promise of the staging of exotic and foreign locales. So, to set the play in the court of Cleopatra would have enticed ticket buying audiences into the playhouse.
Another reason to set the opening scenes in Cleopatra's court in Egypt is to establish her as the dominant partner in this relationship. There are hints in the play that she has Antony under some sort of "love spell" -- again, a testament to the exotic/foreign background of Cleopatra. But, whether he is held against his will by magic or simply by love, Antony is, in many ways under the thumb of Cleopatra, a spoiled queen, used to having her way in all things.
The exotic-ness of the location and the power Cleopatra holds over Antony are two reasons that Egypt dominates Act I.