I can't help at all with info on natural areas, but the driving route from Budapest to Sopron takes you very close to Győr, a pretty city with a larger historic district than Sopron has. Győr also has quite a lot of smallish, art-related museums. I enjoyed them all.
Not too far from Győr (I'd guess about 20 miles) is Pannonhalma Abbey, one of Hungary's key sights. I ran into problems in my attempt to see the abbey; it was a weekend day and there was an important religious event taking place, leading to a very late opening of the abbey. So I'd recommend checking the abbey's website and trying to visit on a weekday when it is going to be open. I'm not sure it's safe to depend on special-event information from the Győr tourist office, because no one warned me about going on the day I chose, and I'm guessing that special events are likelier to occur on weekends. (But that is just my guess.)
My lodging in Sopron suggested I have dinner at the Feher Rozsa restaurant, noting that the food was good and not overpriced (and I did notice higher prices closer to the center of the historic area). I took that advice and was happy with my meal. It's not a fancy place, though.