Bit complicated by bus, I'm afraid. You need to get a bus into Alicante city and then from there to Torrevieja. Buses from Torrevieja to Los Montesinos are every 3 hours with the last one being at 3.30 pm. Not much scope there, I'm afraid.
Alternatively, a bus to Alicante City from the airport (frequent) and then a bus to Guardamar (about every half hour up to 21.30 - same one that goes to Torrevieja). Then a taxi from Guardamar to Los Montesinos which is the closest town (or from Torrevieja - not much difference in distance, really).
Trouble is, you are then stuck in Los Montesinos. The buses through there are pretty infrequent (like, 3 a day) and then only to Torrevieja one way and Orihuela the other. I think it is one early morning, one lunchtime and one 3.30). This used to be the timetable but not sure if it is up to date
Los Montesinos – Torrevieja 7:00 h. – 9:30 h. – 20:40 h.
Torrevieja – Los Montesinos 7:00 h. – 13:00h. – 15:30 h.
Los Montesinos – Orihuela 7:10 h. – 15:40 h.
Orihuela – Los Montesinos 13:15 h. – 20:15 h