Mt. Fuji is a tough but gorgeous hike. It is much shorter than Grand Canyon or Mt. Whitney and can be done on a day trip.
The shortest way to hike Mt. Fuji is to start from the 5th station. I would recommend taking a 2400-yen bus from Tokyo to Kawaguchiko and staying there at night. In the morning, you can start the hike from the 5th station. That’s the furthest one can access via bus. It is about an 8-mile round trip hike with a ~4800 ft elevation gain. So, unless you have done strenuous hikes before, don’t attempt it.
If we had stayed longer, we would have visited the suicide forest and caves in the vicinity.
I, instead, returned to Tokyo and took an overnight bus to Kyoto.
Some people use bullet trains for the Tokyo to Kyoto journey. The biggest problem with the bullet train is the timing. I would recommend against taking the bullet train (JR) in the evening. The last one from Tokyo to Kyoto is around 7 PM. Further, most hotels in Japan don’t allow check-in after 9 PM. So, planning this is a bigger hassle. I would instead recommend taking an overnight bus instead. So that, you reach early morning, in time for your next walking tour.