I had the same thing happen with a Classic once. It also happened to be standing upright, although my logic board was severely damaged. I'm not sure how long the battery had been leaking when I opened the case (I was actually going in to get the analog board from that Mac to try it in another machine to rule out a CRT problem). The analog board and CRT were both salvaged and put to use in another Mac. The rest of it had to be recycled, as the damage was widespread and I didn't want to keep a badly-corroded chassis around. I'm not sure if I was able to save the case or not (I sold a bunch of Classic cases a while back and don't remember which ones were sent off to new homes).

You're lucky you caught it when you did, as it was confined to one area. The entire side of my board was toast.