If this is the iPod that has a touch (not click) wheel, and four separate control touch buttons in a row (between the wheel and the display), that is a 3rd gen iPod from 2003. Although it has a dock connector like more recent iPods, it can only charge using a FireWire port (like the 1st and 2nd gen iPods). It came with a FireWire docking cable.
The cable you brought is a USB docking cable. It can sync over USB, but not charge. If you want to use it with a USB connection, you need a special cable that Apple no long sells. It has a dock connector that splits off into two cables, one with a USB connector and one with a FireWire connector. The USB plug connects to the computer for the data connection. The FireWire plug connects a FireWire power adapter (which also came with the iPod) and supplies power.
I see a used one on eBay
Alternately, you may be able to convert one of the ports on a Mac to FireWire. It needs to be "6-pin" FireWire 400 (the older type of FireWire port) and you'll need a FireWire docking cable.
The best solution is that "two-headed" cable that Apple previously sold, so that Windows users could use iPods back in 2003.
NOTE: iTunes for Windows did not even exist back when these iPods were being sold, and Windows users (who had the "new" USB 2.0 ports) had to use a third-party music jukebox program to sync iPods. Until the 4th gen model, Apple targeted iPods at Mac users. Most Macs had the "high-speed" FireWire ports.
I probably know more about that old iPod than a typical tech at the Apple Store Genius Bar. And they do not sell that special cable anymore at the Apple Store.
If your cousin can find the FireWire docking cable AND the FireWire power adapter that came with that iPod, you should be able to charge it using that power adapter and cable. Then, it may (?) be possible to connect it to your Mac with the battery fully charged, and do a data sync using your USB docking cable.