Unfortunately, DAMAS.es the main bus operator on the Seville - Ayamonte route currently have their timetables off-line.
According to the TUSSAM site, there are buses from Seville Airport to Seville Sta. Justa station at 23:15,23:45,00:15 and 00:45. Sundays and Holidays last bus is at 23:45.
Once you get to Sta. Justa station you then have to get a local bus or taxi to the Plaza das Armas Bus Station. You will probably not arrive there till midnight at least.
From memory and a quick check of other websites, including Cubsur's www.algarvebus.info, there are no buses between Seville and Ayamonte between 00:00 and 04:00. The 04:00 departure is operated by Socibus and is the continuation of the Madrid Service. However, the Socibus site suggests that the service, while stopping in Seville does not pick up for Ayamonte.
Alsa.es has a departure at 23:59 to Ayamonte arriving 01:50.
You should check the availability of this service as it is an international service continuing to Faro in the Algarve and may be limited in winter. It will be a very tight connection unless you take a cab from Seville Airport to Plaza das Armas, which will cost +- € 30 methinks.
Their contact is:
ALSA Customer service
C/ Miguel Fleta, 4 - 2º
28037 - Madrid (Spain)
Telephone: 91 327 05 40
E-mail: alsa@alsa.es
Bus is the only way to get from Seville to Ayamonte by public transport. There is no train service.
My gut instinct is that such a connection will be very tight and that if your flight is delayed you will be in deep doggy-doo.
Better to overnight in Seville and take a bus the next morning from Plaza das Armas. Subject to DAMAS timetable revisions buses leave Seville for Ayamonte at:
0930, 1100, 1230, 1530, 1800 and 1900