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Try doing a "reset to factory settings" and it should take you to the setup screen. I'm not sure what the point of upgrading would be if you wanted to restore from backup. It seems restore with the new iOS would have been a better option from the get go. Anyway, I think my earlier solution should work.
 
In iTunes right click on your phone on left side. Select restore from backup. It won't restore os just all your settings, apps, etc from your last backup
 
The other option is to make a backup of your phone in iTunes, then alt-click Update instead of alt-click Restore. That'll preserve all your data whilst upgrading to iOS 6.
 
The other option is to make a backup of your phone in iTunes, then alt-click Update instead of alt-click Restore. That'll preserve all your data whilst upgrading to iOS 6.

Can you do that? With iOS5 a restore was needed with every beta.
 
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