Greyhound offers a twice daily service from Cairns City to Mission Beach (Wongaling Beach) and Return (the bus service ends Brisbane if you want to head further south.) It takes about 2 hours. Check their timteable on their web page www.greyhound.com.au. Approximate times are depart Cairns 8:15am/1:45pm, depart Mission Beach 10:20 am/6:20pm. There are plenty of Taxis in Cairns City to take you to/from the airport and costs about $20.
Premier Transport offers a once a day service from Cairns City like Greyhound. Check their timteable on their web page www.premierms.com.au. Approximate times are depart Cairns 7.25am, depart Mission Beach 5:20pm.
Mission Link is a daily service Monday to Friday from Cairns Airport to Mission Beach. I takes about 2.75 hours. Check their timteable on www.missionlink.com.au. Approximate times are depart Cairns Airport 2:30pm, depart Mission Beach 6:15 am.