For a contrary view, I have a friend who works in the Sinai (11 years and counting) and has never been to Cairo.
I get her reasoning, if you are going to 'do' the pyramids I think you have to give them time, a day trip from anywhere is pretty barking, never mind Cyprus.
Been to Egypt 12 times myself.... one day I will get to Cairo, and I'll certainly plan to stay for more than a day.
Have you considered the 3 day cruise from Limassol, it is a nice leisurely break good cabins, 2 nights at sea, one whole day in Cairo, and good food. Costs approx �230 for 1 person but well worth it for a little bit of luxury. Ships sail twice weekly.