501 South Texas Avenue
The Wienerschnitzel (called "Der Wienerschnitzel" in the early days) on Texas Avenue is definitely a relic. That is a good thing: there are simply too few A-frame Wienerschnitzels left in the wild. I don't know this particular restaurant's opening, but I saw it in local newspapers (The Eagle, of course) back in the early 1970s, and this type was no longer built after 1973. There's no interior seating (only a few concrete seating benches inside), the drive-through is low and narrow, the storeroom is accessed on the other side of the store (they load food supplies in the second level), and it's slightly overpriced for a hot dog, but it's AWESOME. I even got a shot of the kitchen when no one was working there.
Here's the only photo that's not mine, a photo by Greg D. of Yelp:
There used to be a Wienerschnitzel in College Station, near Walmart but it no longer exists.
EDIT July 8: It is quite possible that this is the oldest continuously operating restaurant in Bryan, but there's Pepe's, and Martin's, operating since maybe the 1940s. Although since Martin's is sometimes closed seasonally, I guess it could arguably be Pepe's.
EDIT May 12, 2013 - Updated to new format, removed then-current preview materials, one of which alluded to the Northgate post, and a "barrage of material for the fall", which amounted to about a dozen posts in varying quality.