Twenty mins away from Shinjuku (on the Yamanote line) lies Yanaka, a quiet part of Tokyo full of temples, shrines and a large cemetary. Quite a few museums and galleries are in the area as well.
I read somewhere this is what Tokyo was like in the early 1900s.

A very charming place, you can only see it by walking around. I did get lost a bit (not following directions on the map). Nevertheless, it was a great way to spend an afternoon in a laidback way.