Your right in what you say here.
you need an APK file. a pay for version is unlikely to be available to download directly, but a lot of free or opensource versions will have apk files for direct downlaod - especially as people like you become more common.
type the name of your given program and "apk" into google, eg:
fbreader apk
gives This page for instance, links to the FBReader apk:-
http://www.fbreader.org/FBReaderJ/
which is here
http://www.fbreader.org/FBReaderJ/FBReaderJ-0.7.8.apk
your device will probably need a file manager to allow you to "click" on the file to install it. I assume it has?
Anyhow, try the FBReader, I use it and it's fine
Jeff