Trips to Cairo can be arranged by yourself or by using one of the local Tour Operators in Sharm el Sheikh.
The cheapest way to arrange this is to get on the bus, either East Delta or SuperJet, and then get a taxi at the other end. The bus ticket will cost you about 55LE. Cairo is over 500Km away from Sharm-el-Sheikh and the road journey will take about 6 hours in the bus and a bit quicker by car.
It is possible to get a hire car with chauffeur to take you. The advantage of this is that the driver will also take you to the sites you want to go to in Cairo. If there are 2 or 3 of you wanting to go then this is a good option   Local operators can arrange a trip either by one of their buses or maybe by plane. Have a look on the Tour Operators Section for some more