You can see Mt Fuji on the shinkansen from Tokyo to Kyoto / Osaka.
There's a Japan Rail Pass for foreigners, gets you on all the trains apart from Nozomi (echo) Shinkansen, which is the rapid limited stop shinkansen.
Normal shinkansen still clocks in at around 280 kph, so you don't miss much.
This pass is useful if you're planning on doing some travelling around, but not worthwhile if you're only doing a couple of trips.
There's coach trips to Fuji, not so easy to get to by train, just because of the network and it being a few changes from Tokyo on local trains.
Everyone's "must sees" are going to be different.
I'd take in Kamakura (old capital) and like a mini Kyoto, but not great if you don't like temples (and you're doing Kyoto). Easy to get to from Tokyo, has a beach in town as well if the weather's nice.

Also next door to Enoshima, which I love. Another beach town, a bit like Japan's California. Nice and slow, but a bit punky and surfer. you can walk out to the island and see Fuji from the sea-candle lighthouse. But again, this is me, I like it slow and not touristy.



Tokyo, Team Labs Planets is a must see, you can get tickets in English online, can't buy at the venue.
https://www.teamlab.art/e/planets/
First week of May is a full week of bank holidays, so Tokyo will be quiet(ish) and tourist spots rammed.