Hi there and welcome
You can get this via iTunes, it is tricky but if you do it properly it willw ork (you though have to do this on a computer the iPod was connected to)
The procedure is as follows
In iTunes, if you don't have the menu bar showing, press CTRL + B.
Ensure you have something to write the number serial onto (pen / paper)
When the menu bar pops up, hold down ctrl and click Help and then About iTunes (hold down the ctrl button the entire time).
A small screen will pop up with some scrolling information, a couple seconds in you'll see 'iPod S/N:'
Write down the list of letters and numbers that comes after that, as that is the serial number of the last iPod that was plugged and synced with iTunes.
You can repeat this process several times to be sure you get all the characters.
I can also do these steps for you, and have made a request via Additional Services. This is optional, and if you choose to do this, I can, with your permission, remote into your computer while you watch and do all these steps (and deal with any problems that arise during them) for you. Or if you do not choose to take this service, we can continue here. This is no problem at all. Just let me know please.
Else let me know please