NO ...no SYNCNo sync needed, u. You need to download the latest version of the app ON PCin the computer so it goes in the cloud and doesn't show as update when u acces appstoreyou access the App Store on iphone ...( this will make 1 badge disapear ) (just use the same apple account)iPhone.
The other badge is from the app allreadyalready installed on the phone ... and thereThere is no way to remove that without deleting iTunesMetadata.plist (iI think uyou can accesaccess the phone ROOT FILESYSTEMphone's root filesystem even without jailbreak - check youtube) ( anyway why do u keep using iOs 6 if u don't jailbreak ... i don't seeother applications on the point computer).