use a Bore Snake to clean the bore.
part of the reason you clean from the breech instead of the muzzle is that you don't want to ding or damage the crown, as this can affect accuracy if it's damaged. there are some types of rifles that are much easier to clean from the muzzle, like lever actions and some semi-auto rifles. i clean my lever actions from the muzzle, but use a muzzle guard to protect the crown. most places that have full line of gun cleaning products should have them. i got mine from MidwayUSA. they are not very expensive.