Meiji Jingu Shrine (明治神宮)
The most famous Shrine in Tokyo middle of yoyogi park with is huge.
Top of Mt. Takao (高尾山 山頂)
If you dont have time to visit Mount Fuji, Mount Takao is easy to visit by train and offers great hiking, views (on clear days you can see Fuji) and plenty of temples. Sacred place for Japanese.
Beautiful large park next to Shinjuku station. Main viewing place for cherry blossoms. When we visited only first trees were in bloom.
Robot Restaurant (ロボットレストラン)
Did not visit the famous restaurant but the Kabukicho (red light district) is unique.
Loved it! You learn the history of samurai in time efficient yet interesting setting! And affordable.