You can download the new version's APK and just move the save files from the old APK to the new one using something like File Manager. That's how most people tend to handle updates for Android; hope this helps!
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You don't need to move save files - the game will do that for you automatically - and you can't move anything to or from an APK as that's just the installer :)
Install the update using the APK, click on "update" when it asks, and then you can delete the APKs - they've served their purpose. That's it! No need to shuffle files around (unless something goes horribly, terribly wrong).