The Settings > General > Software Update option only appears when you have iOS 5+ installed on a device, if you have iOS 4 as your tagline suggests then you can only potentially update to a higher version via your computer's iTunes.
To update, connect the iPod to your computer's iTunes and copy any purchases off the iPod to your computer via File > Devices > Transfer Purchases. You may also want to copy photosand any important documents off the iPod as well e.g. via the file sharingsection at the bottom of the device's apps tab when connected to iTunes, via wifi, email, dropbox etc - they should be included in the backup, but it's best to havea copy of them outside of the backup just in case. You can then force a backup of the iPod via File > Devices > Back Up.
Then start the update by selecting the iPod on the left-hand sidebar (you can enable the sidebar via option-command-S on a Mac, control-S on a PC), and on the Summary tab on the right-hand side clicking the Check For Updates button
Updating to iOS 5+ : http://support.apple.com/kb/HT4972
You can only update a device to the highest iOS version that it supports, which for a third gen iPod Touch is 5.1.1, fourth gen is 6.1.5, and fifth gen is 7.0.4 - I assume that yours is a fourth gen if it's an 8 gig model and it's currently on iOS 4.3.1
If yours is a fourth gen and can only be updated to 6.1.5, then if any of the apps that you want now require iOS 7 then you might be able to download previous versions of them. If the app is free you could try downloading the current version of it on your computer's iTunes so that the app is in your purchase history, and then on your iPod go to the Purchased tab in the App Store app and see if you can download an older compatible version of it