As said, some steel wool or a wire brush will get the rust off with a proper cleaner. Hell, put that gun cleaner to use and use that.
Looks like you are missing the plastic tube for the powder hopper as well as the spring there. I'd bet you may be missing the primer feed tray and/or primer feed spring. That press has 2 primer slides (small and large). It looks like you have the primer explosion shield, there is also the primer tube which goes inside that shield. Again theres 2, a large and a small. As well as the primer tube feeder, which is itself just another long tube with a clip on one end and a plastic little primer pickup piece on the other.
You have 2 shell plates but you will need to see if they have the number on them. As I recall some Hornady shell plates are not labeled. But one will be the icnluded shell plate, and you have the case for the other. I dont hav the bullet feeder so no idea whats needed there.
Anyways I'm happy to go get some pics of my LNL AP for you if you want to see how its put together and what you might be missing.
Nice score.