Terminal 2 Station doesn't close at 8pm, only the JR East Travel Service Center (which is like a JR East travel agency). The regular ticketing office is open until the station closes in the evening. The staff at the airport station is well experienced enough to get you the stuff you want to get to Shinjuku Station.

The current timetable has a 08:46pm NEX to Shinjuku arriving at 10:11pm.
If your flight arrive at 8pm you might have to really rush to get the 8:44pm train, not sure what the practicality of that would be.

The next train doesn't leave until 9:46pm and arrives at Shinjuku Station at 11:04pm.

Any case if you want to get to Shinjuku Station before 11pm and miss the 8:44pm NEX, your next option would be skip taking JR trains and take the Keisei Railways Skyliner to Nippori, then change to the Yamanote Line to Shinjuku Station.

There's a Keisei Skyliner at 9:11pm and 9:41pm. Along with a transfer at Nippori Station in Tokyo will get you into Shinjuku before 11pm.