The most obvious place to go for something like this is probably the county courthouse, but as it turns out I'm told they can only trace who owned the land. If the building was built by someone other than the land owner (e.g., a building contractor), the courthouse won't necessarily have record of that. Nonetheless, it can be a useful starting point. The land owner may even still have living relatives in the area, for instance, who might remember something.

Aside from that, building a building is the sort of thing that generally leaves a paper trail. If nothing else, there should be a building permit somewhere (possibly at the city building). If you want to find that, of course, it would probably help to know when the building was constructed...

Which brings us around to the historical society. If you can find people who have lived in the area for a long time and have an interest in such things, they may be able to help you narrow down when the building went up. If you need to narrow that down further, you can try looking through the archives of the local paper (which the library has on microfilm in the Ohio Room). If there was an article about the building around the time it was built or opened, it may contain useful information, such as ground-breaking dates. (You could potentially even get lucky and find the name of the contractor.)

There are less obvious avenues that can be explored, as well. For instance, if you do a title search and find that the assessment value of a piece of land suddenly jumped to a much larger amount from one year to the next, it may be because a new building was constructed on the property.