There is no direct service.
There is a very good express bus service every two hours between Alicante Airport and Torrevieja bus station. It takes 50 minutes. Can't remember the exact price - around 6 euros, I think.
You can catch another bus from Torrevieja bus station to Los Alcázares. It takes over an hour. It drops off at the bus stop just outside Consum. Costs around 2.50 euros. Alternatively, there are always plenty of taxis at Torrevieja bus station. Don't know the price, but obviously it's much cheaper than a taxi all the way from Alicante.
I've made the whole trip by bus a few times without any problems.
Check the following timetables at: costaazul.net (I can't post a link).
LÍNEAS Y HORARIOS -> Torrevieja - Aeropuerto de Alicante
LíNEAS Y HORARIOS -> Alicante - Cartegena
Another option is a bus all the way from Alicante bus station to Los Alcázares. You would have to get an airport shuttle or a taxi to Alicante bus station from the airport first.