As far as I know there is no "general" internet access available at PHX. A number of the airport clubs do have access, but it's usually limited to the confines of the club and always requires a password of some sort.

In Terminal 4 (HP, WN, BA) the only access that I know of is in the AWA clubs. I know that Terminal 2 (UA, CO, US, AS) has nothing. I haven't been inside Terminal 3 in a long time, but I'm tempted to say that there's not much in there either.

In the city proper I'm not aware of any internet cafes, but access is available for free at most of the Phoenix public libraries (both via supplied terminal or wireless) and a number of coffee houses provide access (either paid or free). 95% of the Starbucks locations in town have T-Mobile WiFi service and all of the Borders bookstores have it as well.