I once had an apartment in the Zamalek area. You can rent them short-term. There are two ways to do this:

1. go through a broker (simsaar) and for a fee they will show you available apartments. This way will likely be faster but costlier.

2. wander through the areas you like and strike up conversations with the bowwab (doorman) of the building and ask him to show you the available flats or to meet the landlord. This can take more time, of course, but this is what worked for me.

How's your Arabic? What areas are you looking at? If you decide this might be an option for you, feel free to let me know....I've some advice about leases and such.....

Good luck!

Cassandra