Yakushima has a small airport with direct flights from Osaka, Fukuoka and Kagoshima. From Tokyo you have to fly first to Kagoshima and then connect to Yakushima.
JAC is the name of the flight company.
You can make reservations here in English : https://www.jal.co.jp/jp/en/
Yakushima is served by two kinds of sea transport. There is a jetfoil service and two ferry services -Ferry Yakushima 2 and Ferry Hibiscus.
Yakushima Tourism Information
Details by PDF (price and timetable)