How can I add power to a USB socket?
Your circuit idea would work but it would be easier to simply use a powered USB hub in between the computer and the USB device. 

I really doubt that the failure of the USB device on the old Linux computer is due to the power though. It is far more likely that the problem is that the old computer does not have the latest device drivers installed that are needed to support the target device. You did not mention what type of USB device it is but even devices that have standard generic capabilities sometimes have special drivers to support getting them initialized and operational.