Molly Malones is near your hotel and they use to have free wifi (last year - so still might be the same)
Starbucks and Holiday Inn also have wifi access.
Theres a list of free wireless access places on the link below. Some are near your hotel.
Phuket Free Wireless Internet Wi-fi
Heres the prices for public use at Holiday Inn
Public space and the business centre
Wireless data connection initial fee 200 baht
Business centre
High speed internet access initial fee 50 baht
Wireless connection initial fee 50 baht
Cathy