Q: No appstore icon on old iPod Touch

You need at least iOS 2 to have the App Store. Also check to see if the App Store is hident by Restrictins. Settings>General>Restrictions.

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

To more easily find compatible apps:

iOSSearch - search the iTunes store for compatible apps.

Apple Club - filter apps by iOS version.