If you need to go somewhere in Phuket, think about public transportation.
The last time I had to go to the airport, I wanted to try something new because I only had a small suitcase.
There are two different ones, a local bus and a smart bus, the smart bus costs 100 Bath but it takes you directly to the airport in an air conditioned bus, the local bus who costs 40 Bath and has no air conditioning and is open, it also only takes you to the old town.
I had time so I took both, I took the smart bus to the old town of Phuket and walked 10 minutes to the bus station where I got the smart bus to the airport at the last second.
This trip took about 1 hour and a half, but it is because I had to take two buses, if you only take the smart bus, you will not even feel the trip.