I can't speak for Japan, but it's dead easy to get work in China and Korea. If you're presentable and don't come off as a "backpacker" you'll probably have no problem finding something in a few days.
I would start with the major cities, pick up a copy of the local English newspaper, and start hitting up schools. Have clean, well written CV's/resumes without typos. Get a local cell phone and give it to the schools.
You and your friend may be able to find work at the same school, but it's more likely you'll get jobs at different spots.
You definitely don't need to know the native language. If you're teaching super young kids, the schools will usually provide a local assistant to keep things from going too crazy.
You can probably do a 6 month contract. Just be up front.
The pay will enable you to live well by local standards, and if you're reasonably smart with your money you'll save a good bit in Korea (not so in China).