There is no water shortage on a small island like Japan, it gets a LOT of rain, and really, when grown in the mountain areas (cmmon in Japan) the rice terraces probably slow the water flow down rather than speed it up (going via the river straight into the sea) somewhat watering the rice crops and all other life nearby.

Rice doesn't actually need a lot of water to grow, they use the water as a type of weed control, rice is unique in that it can grow practically submersed in water. Most other species cannot.

Bali has been experimenting growing rice without such large amounts of water, it went well. Google it!