Hello,
Heat exhaustion can cause laboured breathing or panting and I think they can struggle more with heat as they get older. If you can, keep his cage in the coolest area of the house, avoid handling him when it's hot and put ice packs on or around one part of his cage.
2 years is around the expected lifespan for a Syrian hamster so he's done very well. They rarely live to 3 and a few make it to 2.5 but are usually very frail by that time.
His breathing does seem unusually laboured even when he's asleep. Also I don't know if it's just the way he happened to fall asleep but his head is propped up and sometimes they will do that when they're having breathing difficulties especially from congestive heart failure causing fluid in their lungs. Can you check his testicles and his lips/mouth and see if they have any blue or grey tint? That's another sign of breathing issues. Breathing with the mouth open would be a very late sign of severe breathing problems. Also any clicking/other breathing noises including "wet" breathing noises.
This is a video of my gerbil who had congestive heart failure breathing for you to compare:
DSCN0431 - YouTube. She initially had some relief from diuretic treatment.
Breathing problems are quite distressing for the animal so if you are at all able to get him to a vet, I would. They might be able to treat him and give him a few more weeks or even months, or at the least, if he does have something seriously wrong that can't be treated they can at least give him a peaceful death.