OK, you're biased. If I only had a weekend, I'd skip San Antonio. YMMV.

But if you do need more things to do in SAT, I'm fond of the Spanish missions there.

Also, you should have no trouble walking from the Capitol to the State History Museum to the LBJ Library. This walk will take you through the boring state office complex and the less beautiful parts of the university. It will be mighty hot too, so bring some water with you. But if you're going on 4 September, remember that it's the opening day of Texas Football. In that case I would definitely the aforementioned venues in advance for their parking advice, as football fans will take up almost all of the parking in the northern downtown/university area.