If you're looking for a basic, reasonably priced place to stay on Yakushima, check out Garappa to Kenkichi Minshuku. The main reason I would recommend this minshuku is the staff. They were very friendly and helpful and there is a great atmosphere about the place.
They advised the best times to head off on our hiking adventures and even gave us a couple of car rides to the supermarket and starting points of our treks for a small price. If you're staying on Yakushima for more than a couple of days, it might be worth looking at renting a car as everything is quite spread out. However, I managed just fine using taxis, an occasional bus and lifts from the minshuku staff.
The shower room is very clean and like heaven after a long day walking through the woods. They also have laundry facilities. One thing to bear in mind though, there is only one shower, so you may have to wait a while to use it. It's best to jump in when you get the chance.
The rooms are quite basic. I stayed in one of the western style rooms, which was actually one of several annexed rooms outside the main building. There was more than enough space for three of us, with two bunk beds and a futon to sleep on the floor. Again clean and comfortable enough. They also have Japanese style tatami rooms in the main building.
In the evening, you can hang out in the main room of the minshuku, which had a hostel common room or living room feel. You can share stories of the day's hiking or just relax and watch TV.
They have 100 bottles of shouchu available to buy by the glass. In Kagoshima Prefecture most shouchu is made from sweet potatoes. For me it was an acquired taste, but they have some good local Yakushima stuff included in their selection here. You can also buy various soft drinks and alcoholic beverages including draft beer.
They have a basic food menu too, including rice balls, tofu, sausages, cuppa noodles and other small items. The minshuku can arrange bento lunch boxes to take on hikes, I took them up on the offer...very convenient.
The nearest port is Anbo, if possible try and get a boat to there. I took a boat to the Miyanoura Port, which was fine, but it's about a 30-40 minute drive, which was around 5000 yen in a taxi. The high speed boat that takes a couple of hours from Kagoshima goes to both ports, but there are fewer boats running to Anbo.
I recommend going to Yakushima, it's one of the best places I've been in Japan. If you do, think about staying at Garappa to Kenkichi Minshuku, I have very fond memories of the place.