Q: Ipod does not show software update

To update

The Settings>General>Software Update comes with iOS 5 and later.

Connect the iPod to your computer and update via iTunes as far as your iPod model allows

iOS: How to update your iPhone, iPad, or iPod touch

A 1G iPod can go to iOS 2.2 via iTunes and iOS 3.1.3 via

Purchasing iOS 3.1 Software Update for iPod touch (1st generation)       

https://buy.itunes.apple.com/WebObjects/MZFinance.woa/wa/touchLegacyLandingPage

A 2G to 4.2.1. Requires iTunes version 10.X. If a Mac it requires OSX 10.5.8 or later. With OSX 10.4.11 you can only go to iTunes 9.X which gets you as high as iOS 4.1

A 3G to 5.1.1  Requires iTunes version 10.5 or later

A 4G to 6.1.3  Requires iTunes version 10.7 or later. For a Mac, that requires a Mac with OSX 10.6.8 or later

A 5G to the current iOS 7.Requires iTunes 11.1 if update via itunes

Identifying iPod models