As mentioned above, being in good physical condition is a good start, but even if you are not in the best shape, mountain hunting can still be done, just slower and more carefully/thoughtfully. Tackle smaller hills and mountains until you become conditioned, or shorten the length of the trails taken in the beginning. Just be honest with yourself and know your limits, and then stick within them. While you can push yourself a little more, it will take longer to recover, but overdoing it can be hazardous e.g., heart attack from over exertion. There is no better conditioning for mountain hunting, then hiking hills and smaller mountains with a pack on. Start short and light and gradually work up to your comfort level that will need for the planned hunt.
Others have mentioned your level of determination. Your mental preparedness is as important, if not more than your physical preparations. Be prepared for exertion, pain and weariness. Mountain hunting is strenuous, both physically and mentally. Again, being honest with yourself is most important.
Can you, and are you willing to, put in the long days, the effort and will to see you up the mountain, and then back down again safely?
Are you willing and capable of spending the night on the mountain away from a comfortable camp? (Do you have a light, portable shelter/tarp, pad, space blanket, and water, food and firemaking materials?)
And going down hill, perhaps (and hopefully) with a heavier pack, is actually harder than going up!
And do not worry about keeping up with the younger guys and gals...go at your own pace. Arriving at the destination, out of breath,and over exerted, does not help as it will take you longer to regain your breathing and heartrate so that you can make a good shot on the game, or you'll be so shaky that you'll make a bad shot and miss, or worse, wound and potentially lose the animal. So do not let them over pressure you, as you will be the hunter, not them. I have heard the tales of my Dad, and other guides (both family and non) of the older hunter that was too slow, or when they were lining up on the animal, that talked about wondering why their hunter was not shooting, only to learn that the hunter had died of a heart attack while lining up for shot next to them! It has happened on more than one occasion (I can think of at least 3 of these stories, and have read of a couple more since).
If it is meant to be, and you hunt smarter, you will still have the opportunity to take mountain game. If not, that animal wasn't meant to be yours in the first place. But do not be discouraged! I have gone up the mountain after a specific animal, and did not get the hoped for opportunity for it, and in the end was successful on another animal when the opportunity presented itself. So it does, and ccn happen!
I am in my late 50s and have worked too many years from a vehicle and now a desk, and my physical condition is letting me know that I can no longer do what I could even 5 years ago. And bad injuries to my ankle and knee from my youth are really starting to show now.
I use the pain and swelling meds more than I used to, and have started using my knee and ankle braces again. I have also gotten a pair of a walking sticks.
Mentally, I know that I can still do it, but physically, I will have to practice more of the aforementioned to ensure that I do it safely, at a slower pace.
In the end, the memories and the pride and joy in just being there and experiencing the mountains and all they have to offer is worth all the blood, sweat, and tears! Remember to take those moments of rest on each leg of the journey to take it all in and enjoy being there! (And to rehydrate and energize your body, and mind!)
Best of luck in your quest!
May you enjoy some grand adventures!