Whilst in Tokyo on a Saturday a few years ago, I took the Tokaido-Sanyo Shinkansen to Odawara, then after a local connection to Hakone-Yumoto, the scenic
Hakone-Tozan mountain railway. From there, you can get the Hakone Tozan cable car (actually a funicular railway) to Sounzan, from where you can get the "ropeway" (what we'd consider a cable car) down the mountain to Togendai, a historic port. Sounzan gives excellent views of Mount Fuji, a visitor's centre, museum, tourist amenities and some sulphurous springs. You can get an all-in ticket for the lot, and the Shinkansen will cost you about £20 each way.
https://www.hakonenavi.jp/international/en/transportation/hakone-tozan
Only thing is, it's only 35 minutes on the Shinkansen to Odawara, but it's nevertheless an excellent all round railway day out!
Kyoto is also wonderful, is just over 2 hours on the Nozomi Shinkansen and is £68 each way.