Lots of good advice so far, so I'll try to be brief with a few additional bits...

I think the first order of business will be to find out exactly where the rifle is at now in terms of accuracy before you start spending money and changing anything. Obviously a scope upgrade is going to be needed and sounds like you already have that planned out, so once you have that done go out and shoot some groups to establish a baseline. The reason I say this is because 1) you're starting with a used rifle with unknown history so who knows how many rounds have been run through it or how well or poorly it was cared for; and 2) if you don't know where you're starting from you'll have no idea what needs addressed or be able to tell if anything you do in the future has made things better or worse.

As for a muzzle brake, the factory sporter barrels are too small in diameter at the muzzle to have threaded for a break, and thus to thread the muzzle the barrel would have to be shortened a good bit to get back to a diameter large enough to be able to thread. This would be counter-productive on a 7mm Mag. That basically leaves you two options: a heavier contour replacement barrel that really wouldn't be ideal for lugging around while hunting in the vastness of NM on spot and stalk hunts, or a clamp-on brake like those made by Whitt Machine. Personally I'm not a fan of clamp-on brakes, but many folks here have used them to get the desired result with good success and it negates the need for a heavier barrel or shortening up your existing barrel to have it threaded.

If this rifle still wears the original "tupperware" synthetic stock, I would recommend that be one of the very first things you look at upgrading. These stocks are literally made from recycled milk jugs and are notorious for having a ton of flex in both the fore-arm and wrist areas. Probably the cheapest upgrade option out there would be a factory take-off wood or laminate long-action stock, or a newer AccuStock. A Boyds replacement would be another budget friendly option but a bit more money. The aluminum chassis style stocks are all the rage these days, though they're heavy and not really what I would suggest for a back country hunting rifle. They're also all setup for a heavy barrel so your little sporter barrel would look quite odd in that large barrel channel.