For many people travelling to Yakushima, it's just not possible to arrive in Kagoshima and jump on a ferry the same day. You therefore have to stay the night somewhere, so what are the choices.
Ideally if you have an early start, it's best to stay somewhere near the Yakushima ferry/jetfoil port. These places are all minutes away.
This hostel is on the main road near Yoshinoya and a stone's throw from the port. We've stayed here and it has cheap dorms and small rooms to rent. It's acceptable for 1 night if you're travelling alone. You can book directly in English.
This hostel is a short distance to the port and has cheap dorms and small rooms to rent. We tried to stay here but they never returned our email. Perhaps you'll have better luck.
This hostel is near Kagoshima Chuo Station and has the cheapest dorms and single/twin rooms to rent. You can book directly in English from their website.
These rental apartments are literally across the road from the port. You can book directly from their website or find them on booking.com.
This quality chain business hotel is close to the port and if you are travelling in a pair, might be better value than the low-budget options. We stayed here and thought it was good value for money. You can book direct or via any hotel-finder website.
This large business hotel is in the central Tenmonkan district of Kagoshima. It has a Family Mart convenience store in the lobby and everything you might need for the night. We stayed here and enjoyed their shampoo bar and free drinks. You can book direct or via any hotel-finder website. If you're a couple it may be cheaper than those on the budget list.