I'd like to know about the rust issue as well. Being the tard that I am, I managed to forget that I put my 870 in a leather sheepskin case for two weeks and now there's rust spots on one side of it.
Great stuff! Don't let the fact that it has 'steel wool' in the name turn you off to it. This stuff is exceptionally fine. And yes, a light touch is all you need. Work at it slowly and gradually. A good penetrating oil, ( like Hoppes #9 ) would be good. Or, just go out and get a new barrel for it! Hell, you got a free shot gun, why not? Congrats on the new, ( to you ) shotty!
For hard-to-clean spots with baked on crud or small rust spots, my wife suggested I use those dish sponges with the sponge side and scrubby side. We have the blue ones for use with flat-top cooking ranges and are non-scratch. Works great with a few drops of Hoppe's #9 and a few minutes of scrubbing. The scrubby side isn't terribly efficient, but it doesn't leave scratches or swirl marks on my guns, and you can cut the sponge to fit the shape of what you're cleaning. One sponge generally gives me 9-12 sponge pieces.
if its just surface corrosion, get some very fine steel wool, put some gun oil on the steel wool, and rub over the rust.