Are there any places where I can take a shower in or near the Austin-Bergstrom International Airport? If so, how much does it cost?
Showers are not available at Austin-Bergstrom International Airport, neither in the Barbara Jordan Terminal or in the South Terminal, nor in the two lounges (United and American).
Depending on your plans, the next option would be checking into a hotel and TripAdvisor lists those nearby the airport, many with free airport shuttle.
Another alternative could be a visit to the YMCA; TownLake is probably closest to the airport, about 15 miles, 25 minutes away, which you could access on a guest pass. See the "Guest Pass" section on this page.