The first time I lived in Japan (arriving in August and settling down as a student in Tokyo), Mt. Fuji first became visible from the roof of my university's library in December, along with Mt. Tsukuba to the northeast and the foothills of the Japan Alps.

It is rarely visible in the summer, and iffy in the spring, due to humidity.

If you take the Shinkansen from Tokyo to Kyoto, you may be able to see Fuji from the north side of the train. If it's visible, it will be visible for quite a long while, perhaps 10 or 15 minutes.

The last time I took that route (in June 2006), I didn't see Mt. Fuji, even though I stopped off to visit clients who are based quite nearby. When I asked which direction Mt. Fuji was, they had to confer among themselves and finally agreed that when it was out, it was visible from a certain window. But the humid haze and air pollution prevented me from getting even a glimpse.

I've taken trips to Japan in April and May, but the most recent ones were confined either to the Tokyo area or the Kansai area, so I can't remember what viewing conditions are like at this time of the year.

I agree with the advice to take the Hakone trip. It's touristy, but you get to see a wide variety of scenery.

Otherwise, if it's mountains you want, take the train from Nagoya to Takayama, and after you've enjoyed the historical and artistic ambience of that city, take a bus over the Japan Alps to Norikuradake (above timberline) and Shin-Shimashima.