I recently bought an old cider mill/press in about the same condition. I rinsed it good with a hose, scrubbed it down with soapy water, and repeated until it was fairly clean. Then I pulled the whole thing apart and cleaned all of the grease off with brake cleaner until it was 100% bone dry and used some steel wool to remove as much surface rust as I can. Then I put new grease on the parts that won't touch anything I drink (the screw, pivot points, gears, etc).
Before using the rig, I run 5 gallons of hot water with sanitizer through the whole thing. This takes a while, but pretty much soaks every part that the cider will come in contact with. We've only done one big batch to get a 5 gallon carboy of hard cider going, and it's been about 3 weeks and is looking/tasting great. No pasturization, no chemicals. Just apples and yeast (and a little sugar).
The cast iron parts are still a little pitted and "rusty" looking, but it doesn't seem to have affected the usefulness. None of the rust gets knocked off by the soft apple chunks, and I have a screen in the funnel that I put in my primary that catched the little flecks of "stuff" that ends up in the cider.
I read somewhere that vegetable oil is the best thing to coat press parts in to prevent rust. Pam works great, just keep a very thin film on there to keep any drips out of your product. I figure that I might get a wee bit of oil in the cider, but it'll float to the top and I can rack out from under it when going to the secondary anyway so I'm not that worried about it. I think that you have to be willing to eat a little dirt to press your own cider. There are just so many places to get contamination that you're going to get something in there you don't want. It'll strain out in the beginning or it'll settle out in the end.