Geography of Japan
Terrain
Honshu is the mainfield
Japan consists of four major islands (Hokkaido, Honshu, Shikoku and Kyushu).Honshu literally means "the mainland" and almost all major Japanese citys are included in this largest island. And luckily, almost all major Japanese mountains are also included there.
Mt.Fuji -- the highest peak
The most famous Japanese mountain might be Mt.Fuji, and the highest point of it marks 3776m(12388ft) high. Mt.Fuji is a UNESCO World Heritage site, and a lot of people visit here in high seasons.The recommended season is from summer to early autumn. In other seasons, you have to visit there at your own risk (You can die with ease in this mountain.). Officially, trails are closed in winter.
Around the Japan Alps = Highland

This Alps is surrounded by highland area which contains most of higher mountains in Japan. This area georogically occupies central Japan. To get to the nearest edge of it, from Tokyo station, it takes about 2 hours by train.
By the way, Tokyo metropolis spreads in Kanto Plains (the biggest one in Japan). So there are no major mountains around central Tokyo. If you have hotels in central Tokyo, it needs at least an hour to get to full-scale mountains.
The water paradise
One of the purpose to visit mountains is the magnificent view from the peak. But I should propose another one -- water. Japan has much rain and it erodes many rivers, lakes and waterfalls. Their appearances are really photogenic.And Japan also has many volcanos. Water and volcanos -- exactly, there are tremendous hot springs in Japanese mountains. It is sheer pleasure to indulge in a hot spring ("onsen" in Japanese) after climing!!
Temparature

It is generally said that the temparature will be down by 0.9 degree through going up 100m high.
For instance, when the temparature in Tokyo city is 30 degree, the peak of Mt.Kumotori 2017m(6617ft) is 18 degree.
So the preferable season to visit Japanese mountains tends to be from April to October, of course it depends on the altitude, districts, and other elements.
It is important to study mountain's outline in advance and to choose suitable outfits.
Reference: Japan Meteorogical Agency
Precipitation

Even if you are equipped with anti-hydro gears, the trekking become more discomfortable in rainy day. So you need to avoid the rainy trekking as much as possible.
In Japan, the high precipitation month tends to be June and October. Each one is called "Tsuyu" and "Akisame" in Japanese.
But unfortunately, the daylight hours is suppose to be the longest in June, and you can enjoy the stunning colorful foliage in October.
So if you plan to visit mountains in these seasons, avoid the rainy day by watching weather forecast TV programmes and using forecast applications.
Reference: Japan Meteorogical Agency