We enjoyed 4 nights at the Dewa in February. We had our own car so can't help with info on hotel shuttle. There are taxis parked just outside if you want to make your own arrangements. Depending on the traffic and the major roadworks en route it will take you about 45 minutes to an hour to get to Patong, Karon 15 minutes further, Kata a bit more.
You will certainly see both sides of Phuket between the small and lovely traditional resort of Nai Yang with its beautiful beach, crystal clear water and dining on the beach, feet in the sand style, and the crowded Patong with packed beach and full on night life. Karon is not quite as full on but is getting there. Just further on Kata is a nicer place to visit.