How are Japan's provinces governed compared to the governing method of U.S. states?
Japan's provincial system is much weaker, that is more centralized, than US states. While the provinces are in charge of administering police, fire and health care, things like transportation networks and such are all run out of the central government. I recall one governor who complained vociferously of not being able to move a bus stop 10 meters.
Japanese provinces are more like US counties than like US states.