Mountain bikes are easy and cheap to rent in Miyanoura and Anbo. If you are so inclined, it takes around 8 hours to circumnavigate the island.
You can rent bikes from the following places. The standard price is from 800 yen per day:
You Shop in Anbo
Yakushima Kanko Center in Miyanoura
Yakushima Youth Hostel in Miyanoura (in English)
Yakushima South Village in Hirauchi (in English)
For more information about Yakushima - check out the one and only Yakumonkey Guide to Yakushima