Bluing, either hot or cold is mostly cosmetic and WILL NOT, by itself, prevent steel from rusting to any great degree. To keep from rusting, blued steel has to be oiled and even then, it's the oil and not the bluing that stops the rust.
Back in the day, iron mortars were painted black. A very easy and inexpensive way to paint the tube and prevent rust is to use good old high temp BBQ paint. That's what the NPS uses on the muzzle and around the area of the touch hole of their 6 pounders at St. Augustine.
You don't really have to plug your mortar when you store it. After it's throughly cleaned, before you put it away just use a swab to put a light coat of a good gun oil in the bore. Just make sure to wipe it dry before the next time you fire it.