The Bus Company has this web-site ->
Costazul Site Ok
There are basically three services:
1) Torrevieja service, with city buses. There is a bus-stop just by the pool of the Hotel you mention (100 meter down) and I think there is a bus every hour.
2) The long distance bus from Cartagena to Alicante. Click on Cartagena on the mentioned web-site. The bus is usually at Punta Prima 10 minutes after the time given for Campoamor. Traffic may delay it severely.
3) The long distance bus from Mil Palmeras to Torrevieja. Click on the Mil Palmeras icon.
And, you can call them on the phone. They do speak English and are friendly! I use the bus-service two or three times a week, and is very happy with it.