First, note that this use ("the" with a singular to denote a class) is pretty old-fashioned. You won't find it very much in contemporary writing, and not at all in speech. The plural "Horses run fast" is much more common now.
And that goes over to the case with the adjective. "The common man" as a class is very formal or literary (I immediately think of Aaron Copland's musical piece Fanfare for the common man).
I'm not sure about "A common man" in that way. I think it is possible, but for me it sets up an expectation that you are going to distinguish a common man from some other kind of man. Again I think the plural (probably "common people" rather than "common men", unless you are specifically talking about only males) is more likely.