It isn’t downtown but the Renaissance Cairo Mirage City offers a ‘meet & greet’ service before you get to immigration, as well as a shuttle bus back to the airport.
I’ve stayed there a number of times this year and always use that service.

Marriot also have the Marriott Cairo Hotel which is more central. Maybe some other FTers will know if they provide a ’meet & greet’ service. Cannot comment on how long you should allow get back out to airport from downtown I’m afraid.