Phuket is not known for cheap public transportation, the taxis are rip offs and even Grab is higher than in BKK the capital city.
But there is a cheaper public bus option. It runs from the airport to Rawai Beach, stopping at Patong, Karon and Kata beach too. It is 100 baht and clean and has good AC.
You can get it from beach to beach which I found a much better option than taxi. The Patong beach to Rawai is at the PEA ( search on map) and TO the airport is it Bangala police box
FOR DETAILED INFORMATION YOU CAN CLICK ON THESE LINKS. VERY USEFUL
BUS TIMETABLE HERE
READ ABOUT THE PUBLIC BUS FROM PATONG TO PHUKET OLD TOWN HERE