Yes, yes and yes.
If your stay in Japan is 7 days or less, you can use the 7-day JR Rail Pass to ride the Narita Express from NRT into Tokyo, the shinkansen (bullet train) from Tokyo to Kyoto, local trains around Kyoto (Nara, Osaka, Himeji, etc), the shinkansen from Kyoto to Tokyo and the Narita Express from Tokyo to NRT on the N'Ex. If you will be there longer than 7 days total, just make sure the 7-day Pass covers both directions on the shinkansen and one or the other Narita Expess legs and the Rail Pass will pencil out to save you money.
If your departure from NRT is in the afternoon, it is quite possible to leave Kyoto in the morning, take the shinkansen to Tokyo (or better yet, Shinagawa Station), transfer to the Narita Express and continue on to NRT in plenty pf time for your flight. If your departure is in the morning, then you would want to move to an airport area hotel for that last night.