I am assuming you mean airport not Alicante town.
Taxi will be in excess of €100.
You can get a bus to Torrevieja and then get one to Los Alcazares from there.
There used to be a bus that went all the way from Cartagena to Alicante that stops in Los Alcazarez but I can't find the details now.
Trawl the web using Google or similar - that's your best bet