iPhone 4 cannot connect to internet after iOS6 update
There are multiple things to check on (in the order you should try them):  


Check your settings on "Cellular Data" as they might disappear with the iOS6 update (that's what happened to me anyway)
Try turning on and off your flightmodus
Reset your Network settings
Try rebooting your iPhone
(Restore your iPhone)


Although the latter isn't the ideal solution, it might solve your issue. But only try this in worst case scenario (since it's pretty radically).  

Anyhow, if neither of above options solved your issue, you're not alone. I recommend you pay a visit to your Apple Store.