In & Around:
One of my favorite Observatory, Bunkyo Ward Office. See shooting location map below.
The building has 26 floors. Observatory floor is on 25. It is free and open 9:00~20:30. Check for more details at official site. If you can not read Japanese language, Wiki might be more helpful than the official site.
It doesn't offer a 360 view but it is facing Shinjuku and on a good weather day, you will get Mt. Fuji just behind Shinjuku skyscrapers. They are a big building blocking your view on the right which is not in this photo. Mt.Fuji usually visible during early morning hours and late evening / sunset hour.
I took this photo via 100 mm on a 1.5 crop sensor.