I don't think there is any way to "do it right" without a welder. Nothing will put the metal back except welding in some new metal.
Now, if you're like me, doing it "okay" is good enough
To keep things from getting any worse, you're going to have to remove all the rust, by sanding or grinding it down. This will likely make the hole larger in the process. Once the rust is completely gone you'll want to paint this bare metal with an anti-rust paint - I recommend POR 15.
So, you've got a nicely painted hole.. If you are set on screwing some metal in, I'd suggest you lay a piece of metal over the hole on theinside - meaning that the piece of metal is on the inside of your door, and you can see the rusted metal from the outside. You could then put some body filler in the hole and level it out some - it would look alot better.
Body filler is very easy to work with. If you can spackle a hole in your wall, or work with play-do, you'll be fine with this stuff.