platforms
The Mosin is not the best tool with which to start shooting "long" distances. They were affordable, and ammo too, and were attractive for that reason. The sights on the Mosin are typical blade and leaf battle sights, with little to no easy adjustment for windage, and little more for elevation. Plus, with the assorted heavy ball ammo (yellow tipped ?) that was so common on the market when Mosin's were really cheap, they kick like mad.
If you are familiar with the AR "on duty" (?), then I suggest, if possible, you acquire and shoot an AR, likely the full size A2 rifle with aperature sights. Your results will be far better most likely, than with the Mosin.
If it just must be the Mosin, I'd start on a generous target like a 10" paper plate, at 100 from the bench, and see about group size and consistency. Some old Mosins are not capable of decent accuracy, some are. My thought is that if the rifle is not capable of less than than a benched 3-4 " group at 100, you are probably working against yourself at shooting more than 300 yds or so. If you are not getting groups of any respectable size at 100, determine if it is you or the rifle. Get another shooter involved and see how they group. If your control shooter does OK, you may need more time on your .22.
The x54r cartridge is serious medicine despite it's age, and if your rifles bore is good, will carry accurately way out there, 600 yds+ is probably possible on a discernable target.