Austin's public transportation is CapMetro.

 It is easy to use and convenient to most destinations in the Austin metro area, including Austin-Bergstrom Intl.

Routes 100 and 350 access the airport. 100 runs straight downtown, every 30 minutes, everyday from at least 6-11.

Most routes run every half hour or less and cost $1.25 for a single ride. A day pass for all local routes is only $2.50, or a week-long pass for $11.25. Routes that make fewer stops will cost a bit more. All fare info may be found here. 

For a higher price, the far northern areas of Austin may be accessed by the MetroRail, running everyday, every hour or less from downtown.