It is very rare to find one-bedroom apartments in Egypt as single occupancy is not common here. Most single Egyptians live with their families until they get married and so the demand is not high. You will probably have to go for a two-bedroom place at least. New Cairo is a very big area that is developing in the desert and, unless you are in a settlement like Rehab, you would need to have your own transport. If you look around the forum for Rehab, you will find that other people have been discussing this area and have left some links.