I have the same problem with my iPhone 5, also using China Unicom (in Shanghai). Went to China Unicom, worthless - except could see that the same sim card worked on 3g in a different iPhone 5.
I'm using an Apple unlocked phone purchased in the US - and after going to the Apple store, they suggested that I restore the phone (set-up as new). I couldn't do it at the store, because I hadn't sync'd in a few days, so went home, backed it up, and restored the phone to factory defaults in iTunes. Turned on 3g - a voila! 3g finally works on internet!
Then, I restore the backup of my phone made earlier - and I'm back to no internet. I am trying again, this time wtih the restore from a iCloud backup. Will post back if I have success - but based on the symptoms, it looks like there is some random setting that is captured in that phone restore which is in conflict with connecting regular Safari in iOS to the 3g Unicom network. Otherwise, there's no reason it would have worked after resetting to factory defaults.
Also, to make it weirder - although I coudln't open any pages in iOS Safari on 3g, I COULD successfully send email over 3g... so it was talking to the network and able to send data. WEIRD!!! And frustrating.