Buses from Seville El Prado via Estacion Sta. Justa are operated by TUSSAM, the local transport authority.
Link to the timetable here: http://goo.gl/RRpJx
Unfortunately the first service on a Sunday is at 06:15 arriving at the Airport at 06:45.
If you are flying with an airline that does online check-ins, you may be able to get away with it. Otherwise its a taxi, i'm afraid.
