Well, I have a suspicion that the Jihad event at the start grants a few armies to Eygpt. On the other hand, having a large/huge city starting off gives you a 3-unit recruitment slot, so it isn't too hard.
However, I kept them out until Richard arrived. You need to handle your ecomany well right from the beginning. Selling map information to anyone brings in an extra thousand (Yes, Eygpt will accept, too). The relitivly cheap troops of the Squires and seargents, combined with the large amount of cavalry you get to start off with, should make making armies fairly cheap.
Be aware assulting Cairo, though- Saladin's bodyguard is stronger than most, and there's always lots of armies about.
And I shall go Softly into the Night Taking my Dreams As will You