Riverwalk area is very walkable, and that would include the Alamo. Two days are plenty if you don't drive.
If you don't already have tickets to SAT, consider
Greyhound Express. Yes, that Greyhound
They recently launched Express service in Texas between the four major cities, and apparently it's quite nice including wifi onboard. Unfortunately Greyhound bus stations are involved, BUT if you purchase far enough in advance, $1 fares are available. I checked, and saw some for mid July.
Unfortunately
Megabus just announced that they will not be using downtown Dallas but Grand Prairie due to some issues, and that location is hard, if not impossible to get to using public transportation
Oh, it's about five hour trip, so quite a bit longer than if you are connecting and already past security, otherwise pretty good if you add up the time to and from the airport plus going through security.
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