There is a 24 hour cafe in the underpass between the arrivals and departures area. Certainly does coffee and some snacks. (You would probably need EGP cash, although you might be able to use USD at a premium exchange rate.)

I have always found it fine to sit in the hall outside the BA ticket office until it opens (doing overnight turnrounds on BA, so a couple of hours longer than you have). A good book and a bit of people watching has passed the time. There are quite a few flights going overnight, so it never gets quiet.

There are, of course, toilet facilities, although they are not 5* hotel standard.

On the two occasions I have done this there was once one other person waiting to pick up a ticket when the BA office opened.