Check the Middle East board: there are a few good links in there to firms who can arrange a driver for the day. (Disclaimer: I went in 2009 so I don't know how much has changed since the Arab Spring.)
Rates for drivers all over Egypt, including Cairo, were quite reasonable IMHO. We paid about $50/day (250 EGP) for a Corolla-type vehicle to do a whole day of Pyramids: Giza, Saqqara, and Dahshur. That was from Central Cairo, so an airport origination might be a little higher but shouldn't be outrageously so. That was driver-only: no guide.
You could also haggle with regular Cairo cabbies outside the arrivals area. In theory, that might be the rock-bottom cheap (and most adventurous) way to do it. I just hated the older-style Cairo cabs and liked having a modern vehicle dedicated to me. For a short eight hours in the country, I'd pay extra for the convenience and peace of mind.
Oh, and skip the carpet factory that your driver will inevitably try to take you to. You may need to be strong about this, as "no trip to Cairo is complete without a trip to a carpet factory." Uh, yeah, it is.
We tipped our guy well and we bought his lunch, but we did NOT waste time at a carpet factory.