I am by no means an expert in Ohio law--yet. I intend to get there.
But, IIRC, preemption applies only to firearms. So the part that tries to restrict the possession of firearms more than State law does would be preempted, but the rest of the law would stand. However, that does not mean that you won't be in for the hassle of your life if the city is not set straight before you try to carry in that town.