When entering Tokyo Bay, does the monster in Tokyo City take damage?
You deal damage to the Monster in Tokyo City, then take control of Tokyo Bay (or Tokyo City if the damaged Monster abandons it).

From the rules, page two far right column:


  If Tokyo Bay is unoccupied but Tokyo City is, the Monster who attacks Tokyo deals damage to the moster controlling it, then takes control of Tokyo Bay - or Tokyo City if the damaged monster abandons Tokyo City.