Well, I recommend that you would make a transfer in Tokyo Stations, which sounds easier than finding your way in the complicated Shinjuku station complex.
Narita Express trains leaving NRT all make a stop at Tokyo Station.
Your luggage is light, so you can make this transfer in less than 10 minutes, I suppose, except when an escalator is so crowded that you have to wait in a long queue.
Tokyo Station of JR has one main building and two underground annexes.
- East Japan Railway: Map of (JR) Tokyo Station http://www.jreast.co.jp/e/stations/e1039.html
In Tokyo Station, Narita Express trains use the tracks on the 5th basement in the Sobu underground annex, which is around the bottom of the map.
(A Narita Express train leaving NRT for two destinations is divided here into two parts.)
You can first go up by escalator to the intermediate hall on the 4th basement then go further up by escalator to the concourse on the 1st basement.
You pass through Marunouchi Underground Central Gate to exit JR, walk on the short pathway to Tokyo Station (M-17) on the Marunouchi Line and get through the gate to enter Tokyo Metro.
Then, you go down by elevator or escalator to the platform and catch a train bound for Ogikubo at Track 1, which should bring you to (M-05) Shin-nakano in around 25 minutes.
- Tokyo Metro: Marunouchi Line timetables: Departures for Shinjuku and Ogikubo from Tokyo Station http://www.tokyometro.jp/lang_en/station/tokyo/timetable/marunouchi/a/...
- Tokyo Subway Route Map (with station codes) http://www.tokyometro.jp/lang_en/station/rosen_en_1702.pdf