There are direct bus routes between Malaga Airport and the Marbella bus station.
The trip takes around 45 minutes and the ticket costs just over 4 euro.
Telephone number to the Marbella bus station for info is +34 952 764 401

In Marbella you can buy tickets at the station. In Malaga Airport tickets are sold on the bus.

Buses go from Malaga Airport to Marbella at: 05.30, 9.15, 11.00, 12.45, 14.45, 16.30, 18.30, 20.15 and 22.15.

Buses go from Marbella to Malaga Airport at : 06.15, 08.00, 10.15, 12.00, 13.45, 15.30, 17.15, 19.30, 21.15, 23.00

Offical Marbella Bus website