It can't really be compared to Penang or Melaka, but Phuket's old town area is definitely worth a look if you're interested in that sort of thing, and looking to spend some time away from the beach. The old town area covers a couple of blocks. It is not well developed as a tourist destination, but many of the buildings are still in very good shape.
Phuket has always been an important trading center. The large, protected natural harbor where Phuket town sits made it an attractive place for traders from the east and west to meet. Development really took off in the late nineteenth century, when the demand for tin skyrocketed in Europe after the invention of the tin can. Merchants in British-occupied Penang soon realized the vast deposits of tin that were available around Phuket and began coming to the island to exploit its resources. They were followed by waves of Chinese immigrants to work the mines. The newcomers also bought with them the architectural models of the colonies, such as the shophouse and five-foot way.
As tin mining gave way to tourism as the island's primary business, Phuket Town became less and less important. That turned out to be a good thing. Since property in the city didn't have nearly the value of the beachfront, there was no incentive to tear down the old buildings and build new ones. Then, just a few years ago, people began to realize what a rich architectural heritage they had on their very doorstep. Buildings were protected, fixed up, and are just beginning to get attention from international travelers.
The Phuket Old Town area is really rather small, and easily navigated on your own. You can follow our suggested walking tour route, or, if you'd like personal tour through the history of old town, check out Phuket Heritage Trails. They operate guided tours of old town, and elsewhere, lead by descendants of the people who built many of the historic places.