I have the same error. Did a permission repair, re-install iTune10.1, still can't resolve the error. I had this error on my Macbook Pro, latest OSX, and currently trying to restore from iMac. Will report soon. very frustrating and waste of time.
I read a post suggesting software to kick the phone out of recovery mode, something like Recboot, but not sure exactly. Try and search, you should find it since it hasn't been that long ago.
OK. To report back, since Apple troubleshooting guide does not help much.
I managed to restore the iPhone using another Mac (iMac). So it must have been my MacBook Pro or the itunes in there that is causing problem.
Try TinyUmbrella to kick it out of recovery. You will find that 4.2.1 was installed and the phone works. Check everything, because it might be a corrupted install.
TinyUmbrella does not help for my error 1013 on iPhone 4, only different iMac helped. I am going to delete and re-install my iTunes on MacBook Pro just in case, since I have a couple more iPhones and an iPad to update.
BTW, anti-virus software would not be the problem cause I have the same installed on the iMac.
You tried another computer, made sure you are using the USB on the computer itself and not a hub, and have any firewall and/or anti-virus software disabled? While a 1013 error generally refers to a hardware issue, many have said they are taking care of it with software that can kick the phone out of recovery.
I'm sorry, I don't have any futher ideas. I saw you posted in another thread where someone was successful in updating despite the same error. I just say you will need to keep watching here for any other possible solutions.