Having spent my entire adult life at (and most of it well above) mile-high altitude I've never had a problem with hunting at or above 10,000' MSL, so I really can't relate to those who find themselves struggling in the high country. I think the advice, such as Aoudad Shooter's tips, of actively increasing your aerobic function while still at sea level is spot on but once at altitude I would advise a very different plan of action.
In the high country try to move as slowly as the hunt will allow. The idea here being to never become breathless in the first place. Moving slowly, and intermittently, when hunting also enhances your likelihood of spotting those sheep before they are alarmed by your presence. Spend at least as much time stationary and glassing your surroundings as you do moving. Your slow and intermittent ambulation, with no shortness of breath nor aching muscles, will also enhance your safety with little risk of tripping, falling or stumbling on the uneven terrain.
Yes, do stay well hydrated but also maintain a healthy level of salt in your diet as the disabling effects of salt depletion are very common among low altitude visitors to the high country. Just a little bit of salted nuts, pretzels, crackers or whatnot in your trail mix can be good insurance against the debilitating onset of salt depletion.
Most of all, have a good, safe and comfortable hunt in our western high country.
P.S. I took the attached photo at the 14,000 foot level, 35 miles northeast of my Colorado home.