By the time you finish checking in to your hotel, it will be around 11:30pm.

Explore that night by walking around the Shinjuku station area, possibly including kabuki-cho. You may get stranded if you go to Shibuya. Public transit is mostly closed down by 12:30am, although the last train from Shibuya to Shinjuku is around 12:45. Otherwise, it's taxi only.