I think ramen makes a very good late night meal. This plece "Hakata Tenjin" is my one of my favorite ramen chain. For 500 yen (about US$4.50), you can get hot bowl of Tonkotsu Ramen. You can ask the cook to cook noodle soft or hard. Any you also get second serving of noodle free, provided you left some soup in your bow.
https://www.tripadvisor.com/Restaurant_Review-g1066457-d3752447-Reviews-Hakata_Tenjin_Shinjuku_West_entrance-Shinjuku_Tokyo_Tokyo_Prefecture_Kanto.html