zagraniczniak:
Yes, there is no play on Sunday on the Old Course (see exceptions). You may walk the course, alone or, as I did, in a tour guided by a caddie. They pull the flags out and place one in an extra hole left to the 18th green, a good opportunity for a picture with the R&A clubhouse behind you.
And, because the areal is public ground, there are pedestrians through the green every day, e.g. walking their dogs. There even is car traffic on the road crossing the 1st and 18th, so beware of your drive!
And, speaking of playing there, the adjacent courses, "New" and "Jubilee" (both built in the 19th century), have a lot of the same spirit with less players, lower prices and no HCP limits (AFAIS no HCP necessary). I played Jubilee on a calm day and enjoyed it very much!