Visiting Shinjuku Gyoen National Garden on a rainy day isn't too bad if you have an umbrella. There are barely any people and its like you have the whole place to yourself. I highly suggest pitching the umbrella and running through the place like Pheobe from Friends.
Our apartment was in Shinjuku so that's where we spent most of our time. Which turned out perfect because it was my favorite part of the city. Theres all these great alley ways, stores and restaurant. Every where you turned, there was something new and interesting to see.