As for cheap hotels, there are some on Broadway just outside downtown that are usually pretty cheap. The El Rancho comes to mind. But mind you, we're talking primitive here.
Things to see. In my book, the Old Spanish Missions are a nice visit, and free. They're scattered in a chain south from downtown.
Brackenridge Park is scenic, in a dilapidated, WPA-built sort of way. It's free, except for attractions like the Zoo (which is good) and the Witte Museum (which is a little youngster-oriented for my taste.)
I'd certainly visit the riverwalk, and probably try to splurge on dinner down there. There are certainly some adequate cafes along the river. My favorite, though, is the Liberty Bar on Josephine street. You have to see it to understand.
If you're into art, the San Antonio Museum of Art is worth the small cost. Also on the art subject is the McNay Art Museum, in a cool old mansion.
The Institute of Texan Cultures in Hemisfair Park is pretty interesting, and right next door to the Tower of the Americas.