The cheapest and most fantastic way to go from Japan to Taiwan is by boat, through the route: Tokyo - Naha (in Okinawa) - Ishigaki - Iriomote - Keelung (in Taiwan).
I stopped for a week in Naha and visited Itoman caves, in the south.
The shipping company was called Arimura Line, and life on board was amusing talking with friendly Japanese. In the nights I was given just a pillow and a blanket and slept on deck.
Until the beginning of the XVII century Okinawa archipelago belonged to China, and then the Japanese invaded it and called Nansei-Shoto. There is still a military US base in the north of Okinawa.