Under 50cc. (Gentsuki)
Dust cheap in insurance and taxes. License comes with your drivers license. No annual inspections. Bikes themselves can be had for next to nothing. Honda Super Cub runs 40km/liter of gas. Unquestionablly the most economical transportation in the world. Good for commuting, pizza delivery and like applications. One person only. Carrying passengers is illegal. Mechanically limited to under 50km/hr. Not legal for highways (in case you are crazy enough to try).
Under 125cc. (Kogata)
Still cheap. Carrying one passenger is legal. Legal for highways. Good compromise.
Under 400cc. (Chuugata)
Most popular. Taxes and insurance cost real money. License does not come easily or cheaply. Inspections required every few years. Best for the recreational biker.
400cc and higher. (Gentei kaijo)
Taxes and insurance will break you. Getting the license takes the endurance of a Marine boot camp. Regular inspections are designed to discourage ownership. Most popular is 750cc. Beyond that, the legal hassel becomes overbearing and the cost prohibiting. You will be universally respected for riding this class. Not the most practical though.
By coincidence Oita, along with Hokkaido, is a popular biker destination. Yamanami highway is a good place to bike. We see a lot of bikes over Golden Week. Depends on the kind of biking you want to do though.
I've been away from biking for so long, I am vague about the specifics. I don't know about how foreign licenses are treated.
Almost nothing gets stolen in Japan but bicycles and motor bikes are exceptions. Be sure you have a good place to store it.