Export and import data from iPhone after restoring with iTunes (or 3rd party apps)
Interesting... What happens when you plug in your iPhone to the computer? What's the message of iTunes? As far as you can find the iPhone from your iTunes, you can always do the backup by right-click on your iPhone icon in iTunes and select "Backup" to force it to back up. 

However there is one thing you should know: based on my experience when upgrading the iOS 5, even I did the force backup as above, after restoring it still not the same as the point when I did the backup. I am not sure whether it really "backups", but at least this is an option.