I went to this Izakaya with my friend on the other day and I loved this place called "Cha Cha Hana" (茶々花)
The Izakaya serves in Kyoto style, which means small Japanese dishes called "Obanzai". You will be able to taste various delicate Japanese food here. They also have a pretty wide selections of sake.
Although they are located close to the famous Golden Gai, you will walk on a narrow street with lots of greens. Feels like entering to a hidden gem in Shinjuku. The atmosphere inside is very nice, too. Definitely recommended!
If you want to go more casual, try hitting Omoide Yokocho or Golden Gai, or Shinjuku Sanchome District.