Just google "spin gym" or whatever for Google. Too many local brands to list these days. Sure they would do a drop in class.

But IMHO, eff that. Go for a run in Umstead state park instead. I'd rather do that any day (even as a cyclist and non triathlete) than ride a spin bike. Either park in the park at the central gravel lot or run in from the NCMA for a longer run. The NCMA has some hills, paved, gravel mix good for running also in off-peak hours. NCMA gets busy for a run mid day on weekends. Umstead runs would be fine.

There's also tons of greenway around to run on, just that Umstead has less of the "off leash dog, off leash child, crowded" vibe to it like greenways can.

So I vote run in Umstead. Choice of wide gravel path routes and more of the trail runner stuff in there.

The American Tobacco Trail is a great place to run. Starts Downtown Durham by the Ballpark and runs south for 22+miles - http://www.triangletrails.org/american-tobacco-trail

Another vote for Umstead. Trails (and shade) for miles, not too crowded, beautiful place to run. Might want to use some bug spray, but if you run fast enough and don't stop, you may not need it :-).