ASSUMING that you have a good bore, there are a couple of things to look at.
Start with the crown of the muzzle, and the last inch of the muzzle. If the crown has been dinged, dented, or molested, it can have a disproportionate effect on accuracy. Cure is to recrown. This can be done by a smith- or minor dings can be done by you with a round headed brass bolt, a drill, and some lapping compound. The last inch of the bore is sometimes found with very worn rifling- from cleaning rod abuse. A THOROUGH cleaning with a good solvent may help get you some depth back with the rifling (crud sitting in the bottom of the grooves) but for severe wear, it is either counterbore the muzzle (see gunsmith) or rebarrel.
The sights could be better- and further apart. That may not be something you want to fix if you want to keep original configuration.
Two things you CAN work on- reducing lock time, and improving the rather crude trigger. This is a link to a lot of info on these rifles- look at the articles over on the right side. Enjoy- Collecting and Shooting the Mosin-Nagant 1891/30 - HISTORY