Q: whenever I try to install a game I downloaded from iTunes, it's prompting me that unable to install, needs software update

Some apps have minimum requirements, and if your device doesn't meet it then you can't use the app.

Same premise as if you buy a program for your computer that runs on windows 7+ and your PC only runs Vista.....program won't run.

In your case, if you can't update your device, you can't use the game. So you have to find another game or buy a new device.

It appears you have a 1G (original) 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

Identifying iPod models