If you intend to do a complete re-blue, just send it in to a re-finisher and they will remove the rust, polish and re-blue it.
If you want to do a home job, cold blue will not do a very good job.
Cold blues are intended for touching up scratches and small worn spots.
"Usually" when you attempt to do a large area, what you wind up with is a streaked, sort of smoky blue-gray.
In any case, cold blue is NOT durable at all, and wears off very quickly. Cold blue also tends to turn brown and develop rust, even when lubed.
My suggestion, since this is not going to be a show barrel, simply remove the rust and coat it with one of the spray-on gun coatings, or even PAINT.
You can buy various gun coating from Brownell's. Some need to be baked in an oven, others are just sprayed on and allowed to harden.
These come in many colors and are quite durable if applied properly.
You can even use Rust-Oleum oil-based paint.
Simply spray on a heavy coat and bake in the kitchen oven at 300 degrees for an hour.
Baked-on paint is quite durable and lasts a long time.
(Do the baking when the wife is GONE for the day. It smells the place up).
Removing the rust depends on what you're going to refinish with.
If you're going to coat the metal, the best option is to have the barrel bead blasted. This removes the rust and gives the surface a good "tooth" for the coating to bond to.
Most auto garages, brake shops, or machine shops can bead blast it.
If you want to retain as much original finish as possible, buy some Kroil or any other good penetrating fluid.
Liberally coat the metal with the fluid and allow to stand for a day or two to soften the rust.
After soaking, apply more fluid, then use a brass "toothbrush" to liberally scrub the rusty areas.
After brushing, wipe the metal dry, and either coat with CLP Breakfree to prevent more rust, or degrease thoroughly and coat the metal with a finish.
Another method for removing the rust, most pits and ALL the old finish, use strip's of sand cloth "shoe shine" fashion to polish the barrel clean and smooth.
When applying a finish at home, how well you clean and degrease the metal determines how well the finish turn out.