The 2nd generation iPod touch can’t be upgraded to iOS 5.
One PDF reader that still works on pre-iOS 5 devices is GoodReader, although according to their site it requires iOS 4.3.
If you connect your iPod touch to iTunes, then there should be an option to update to iOS 4.3. The instructions in this Apple Support article explain how to update an iOS 4 device (see Update your device using iTunes).
There may be other PDF readers that can run on iOS 4 devices, but I don’t know of any by name. Pre-iOS 5 devices are now a very thin sliver of the iOS installed base, so most developers have dropped support for them.