Cairo airport, along with every site in Egypt remotely connected to tourism, has ridiculous amounts of security, so you are in absolutely no physical danger. If you wait until morning, all that's going to happen is that you'll be stuck in a traffic jam on your way to the city!
What's more, since the revolution, Cairo's rather notorious unmetered taxis are no longer allowed at the airport. If you ignore touts, pick a cab off the rank and insist on the meter, you should be perfectly fine. (Well, aside from the abysmal standards of Egyptian driving, but that's another story. I still remember driving to the airport one night past the wreck of a BMW that had literally wrapped itself around a lamppost.)
All that said, if somebody is arranging a transfer for you, it's presumably not going to be an ordinary taxi, but somebody known to whoever is sending them. So get them to stand outside Customs holding a placard with your name on it, and get their phone number as a backup in case you can't find them.
(Mild disclaimer: I haven't been back to Cairo since they opened the new terminal, but it wasn't a particularly hard airport to deal with even before.)