Hi Harrison E,
Prepare to become addicted
I just assume you are planning your first sheep hunt. If so, I would focus on most of the other gear more than I would on my choice of rifle. That and serious year-round physical training.
I thought I was in great shape before I hunted Rocky Mountain Big Horn in 2013. I am convinced "Sheep Shape" is almost unattainable unless you are climbing around the nastiest mountains you can find most of the year. Probably depends on where you will be hunting, but if your quest for sheep takes you to the Rocky Mountains of the US and Canada, or to Alaska, the primary limiting factor to success will be how good your physical condition is and how good your gear is.
I know you asked specifically about rifle and cartridge selection. Sorry to dodge the question
I'll hit that in a moment
Seriously, I think the best advice I can offer is this; be in great shape! When you think you are in the best shape of your life you're halfway there. Redouble your efforts and train harder!
Other gear - make certain you have the best boots money can buy. I really like Kennetrek's. Well broken in of course. I also like a merino wool or synthetic liner sock under a top quality wool or synthetic heavy boot sock. Make sure you have the best binoculars and spotting scope you can afford. Also, I have the KUIU bino harness that is the best I've tried so far. Very worth the $80 I paid for it.
Make sure your clothing is lightweight layerable all weather synthetics. Alot of sheep hunts take place in locations and time of year where you may have to adapt between temps of 60 degrees and sunny to blizzard and sub-zero. Top quality, lightweight gear is essential, not only for comfort but in some cases survival.
Okay, I hope that doesn't sound like a lecture. Not meant to be. If you got all that covered, perfect! You are way ahead of the game.
On choice of rifle and cartridge, I use a stainless Ruger Hawkeye with a Hogue stock. Not fancy or expensive, but very accurate and fairly lightweight. I pack it all day no problem. It is chambered in .280 Rem which in my opinion is almost perfect for sheep. Sheep are not especially tough animals to kill. Not as tough as a mule deer. A high BC bullet around 150 grains is good sheep medicine out of a 7mm. Another great choice would be a 6.5/284 with around a 130 grain bullet. Any of the WSM's from .270 thru .300 would be great too, although a .300 WSM is more gun than you need. For 500 yard shots you may be glad of the extra punch. The challenge there will be that the wind always seems to be blowing in sheep country and 500 yards is a pretty good poke at a sheep when the wind is blowing. I personally would not plan to shoot that far, but that's just me.
I really like Kimber rifles and if I were buying a brand new sheep rifle and money was not too much of an obstacle I would buy their Mountain Ascent in .280 Ackley Improved. That in my opinion would be about as perfect a sheep rifle as you could get.
Topped with something like a Leupold VX-6 3-18 would make one hell of a sheep rifle!
Regardless of rifle, you will have the experience of a lifetime
Let us know what you decide and I hope to read a successful hunt report from you in the future.