There is a taxi rank outside the arrivals hall and it should cost approx 10-15 euros to get the apartment.
Take a look at On Arrival for more information
Some Notes on Car Hire:
A hire car is recommended and the road network is very good and traffic is very light compared to the UK so don't be put off hiring a car.
When using the car hire search engines and flying into San Javier Airport you need to select San Javier Airport which will sometimes be called Murcia Airport (MJV).