Well I think OP wants to see cherry blossoms, and March 30 is close to the peak of their bloominess, according to forecasts. It's worth staying near Shinjuku Gyoen for that alone, although one would hope that it doesn't rain.

I don't think even the staunchest defenders of CoCo Ichibanya here (...they know who they are) would suggest eating there for your only dinner in Tokyo though. If you want to eat sushi on your first night, Sushi Ten serves dinner until 2am, and is inside the bustling, 21-hour-a-day NeWoman food hall very close to your hotel. Prix-fixe menus are 7000 yen.

Toriyoshi, about five minutes away, is open until 4am, and they serve fantastic chicken wings along with yakitori and good side dishes. Watami is a lively izakaya and it's another late-night option on that side of the station.

There's plenty to see and do in Shinjuku the next day even if it rains - you can compare several different department-store food halls like Takashimaya and Isetan, and check out the new Samurai Museum.