Lucky bugger!
I don't know of any cigar shops in Phuket, but if you go to Otop shopping market at Patong (just up the road from Jungceylon shopping mall), look for a bar called Blondies2. The guy there, Mick, is an Aussie ex pat, and a great guy. He'll look after you. He might even know where a cigar shop is.
Hope you have a great trip mate.
Mitch