One of the maybe less known Secret Escapes [for FREE] is this amazing view. And if you follow the signs from Shinyuku Station you get an extra amazing tunel-walk as a plus.
⇒ Tokyo Metropolitan Government Building [short: 東京都庁, Tōkyō Tochō]
As it is for free, people might line up at the entrance. So my recommendation would be a -sometimes- rainy monday noon. You will have to choose right or left tower, where 2 elevators are waiting for you to bring you up the the 43. floor . [be aware your ears might hurt like mine, if they are not used to such a sudden change of height]
And up there you get the spiderman-feeling, and if lucky even are able to get a glimpse of Mt. Fuji, Yoyogi Park and if lucky you will see some rain fall over on Shinjuku, while its definitely sunny on your side. They also have a café up their and some tourist info/shops as well as free wifi.
Another great thing is the building itself¹. It is huuuge (in my point of european-view) and it makes fun just to walk around inside, outside — I love this non-tourist escapes to the “real”world.
You also could add the “other side” of Shinjuku station, where similar buildings line up, and a huge shopping center awaits you with all it`s curiosities. [Great sunset inclusive]
⇒ Takashimaya Times Square [Shinjuku Bus Station]
¹Sidenote: You might want to google Kenzo Tange, the architect behind this great building