You cannot omit the World Heritage site, Mt Fuji, when it comes to hiking in Japan. It is the highest mountain in Japan, which stands at 3776m above sea level. It has steep slopes occasionally, but relatively easy even for beginners. Many people climb to the hut close to its peak on the first day and catch the sunrise at the highest peak in Japan next day. In addition, there are more popular mountains for a hike in and around Tokyo, such as Mt Takao, Mt Ontake in Tokyo and Tanzawa in Kanagawa. You can find further information on hiking and climbing in Japan on Lonely Planet.