iBooks will not download to it, as it requires OS7,which the iPod 4th Gen cannot accommodate... I downloaded Adobe Reader, but can't locate the file on my iPod for it to open (when it's attached iTunes tells me the file is in "Books", but I can't find "Books" on the iPod itself)
As of version 4.4.1 (current latest version), Documents by Readdle only requires iOS 6, and therefore should work on your device.
The books in the iTunes books section will only sync with iBooks. Simply un-sync that section, since you don't have iBooks installed. Download, install and open ’Documents' on your device, then go to the Apps section, scroll down to the file section, select Documents and browse to your files there, then sync.