As far as I know, it is possible to visit Memphis by public transport but it's difficult and time-consuming. We came as part of a day-trip to Dahshur, Saqqara and Giza, and paid 220LE for the taxi and driver for the whole day, through our hotel.
I decided not to use public transportation to see the different pyramids outside Cairo (Dahshur is not easy to reach by public transport), but to pay for a car with driver. I asked at the travel agent, which is open a few hours a day at Garden City House Hotel. There was no one to chare the price with so finally I went alone. For the transport to Dahshur, Memphis, Saqqara and the pyramids at Giza I paid 150 pounds. All sights are close to Cairo and with a car they can easily be seen in a day.
It is possible to visit Memphis by public transport but it is complicated and takes a very long time to go there. I don’t think it would be worth it.