Shinjuku is filled with night time attractions. You can find bars and Izakayas (Japanese pubs) everywhere.
Omoide Yokocho and Golden Gai are two of the most famous nightlife spots. Both of them have tiny traditional bars and Izakayas.
Shinjuku Sanchome area near Golden Gai is filled with great Izakayas and bars, too. There are many bars open until morning or 24 hours manga cafes so even if you missed the last train, you never get bored for things to do there.