If your shooting Corrosively primed ammo, simply use boiling water poured down the barrel, a very common cleaning stepfor all militarys that have/do use corrosive primeing. The water will dissolve and flush the corrosive primeings salts out of the bore. A good scrub and a rinse with boiling water , the heat of the now clean steel will dry itself, and after that simple step , clean with solvent, patches and a wipe of oil over the bare metal bore when finished.
If your not useing corrosive ammo and the bore is still rusting, its most likely a Humidity problem while stored.
Youll need to wipe a heavy oil down the bore after the cleaning to 'seal' the bare metal bore from the elements.
I usually wipe the oil out with a patch before shooting it.
As for Rust removal, soapy water and a boiling water rinse scrub with a good stiff brush will help get you down to the steel, solvents and copper/lead removeing agants afterward will show you what you got till your patches come out clean..............some rifles have problem spots, and all the scrubbing in the world wont help, but if its accurate, theres not much you can do but use it well
.
I use a 20 gauge brush on the chamber, with a solvent or boiling water to remove any crud, cosmo, corrosion or carbon. This keeps the "Stick Bolt" away that arrives with dirt, laquer, neglect and such getting pounded in the highly pressured and hot chamber till its built up, "sticky" and hard to open. On a brand new rifle I use brake cleaner and free the chamber of all preservative, 60+ year old, dried cosmoline that actually looks like the steel. A clean chamber is a smooth operating Mosin. This is one place I will chuck the rod segmant from my home cleaning kit into a cordless and twirl the chamber from the breech end. After the first chamber scrub, all others are easy.
Power Twirling a brush down your bore will leave you with the sides of the lands uncleaned, so slow strokes down the bore with a rod or tip that rotates with the rifleing is the best method. It will ruin your metal brushes very quickly too.
If your Mosin came with a Soviet cleaning kit, it will have a muzzle cap with a hole, a connection peice to slide over the end of the rod and a small metal rod to make a handle out of. This will allow you to attach a brush and 'pump' the rod back and forth from the muzzle, wipe with patches and oil the bore without touching your muzzles crown, and then a chamber scrub from the breech.
Finnish cleaning kits have an extention to scrub the barrel from the breech.
Anyway you clean it, it must be thorough, not damage the bore or muzzle and some protection for the clean steel and you should have that rifle for the rest of your life.
Good luck and Good shooting.