In all honesty your probably better of getting a flight with Thai air... Bangkok airways.. Nok air... Or maybe even if you really have to Air Asia from Bangkok to Phuket And renting a car when you get into Phuket.... Driving will take you far to long and a waste of time..... Nok air is probably your cheapest option but you will need to watch the weight restrictions i think from reading its 15kg baggage allowence but its probably the cheapest option... Will take about 1hr 40 mins approx by air as opposed to god knows how long by car.... Then get a car in Phuket to rent if you need one.... To be honest the flight with Nok air might even work out cheaper for a return than renting a car from Bangkok
www.nokair.com
hope this helps