Here's a few option for your question.
A personal favorite of mine is this restaurant called Negishi (map). This is a Japanese restaurant famous for their beef tongue dishes. You can also get grilled or fried chicken meat dishes and pork. Prices vary between 1000yen - 3000yen.
For a more traditional Japanese restaurant, you could try these sushi restaurants in the area
Himawari Zushi Shintoshin (map)
Itamae Sushi Nishi-Shinjuku (map)
Here's another option for some udon and ramen restaurant
Ichiran Ramen Shinjuku East - Exit (map) -> this one might be crowded since it is really famous among tourist.
Hope this helps!