No. You'll want to transfer your music to iTunes on the PC otherwise your friend will loose the stuff on the iPhone. The iPhone can only be tied to one device, and cannot sync to more than one. You can transfer purchases from the iPhone to your computer... Note this method will NOT transfer any music etc he didn't get using his account from the Apple Store. To do that you will need other methods.
First he'll need to authorize your computer with his iTunes account, select Store - Authorize this computer.
Then plug in his iPhone, when it appears in iTunes, Select File - Transfer Purchases. iTunes will move all his purchased music and apps from his iPhone to your computer.
I would then make sure it's all there and ready for the update by backing it up again (right-click iPhone in iTunes and select backup).
Now you can download and update like normal (there's a button). All his apps and settings should transfer over and remain intact.
Thing is he's just going to have this problem next time. Why not just use his laptop and go buy a new power cable.
Best of luck.