At the beginning, we see a youthful looking Zod's coup result in failure. Then when he arrives on earth, he has visibly aged complete with greyed goatee. But his followers do not seemed to have aged. Why did only Zod age?
Zod was already elder to rest of his associates and followers when they were exiled.
For e.g. considering equivalent in earth years, assume that the first time we saw him on Krypton, he was 70ish (remember, kryptonians age slower than humans. He could be 70ish and still look that young), and his followers were 30-40 The next we see him during the events of the movie, 33 years have passed (General Zod).
That would make him 103 years old at the least. That would look old even by kryptonian lifespan standards. By comparison, his followers would be 60ish which is comparatively younger considering krptonian standards.
In conclusion, I feel that he looks older than the others only because he already was older than others in the first place. That, plus the responsibility of the entire race upon him, was sufficient to make him look elder than the other survivors.
Zod likely did not age anymore than his fellow insurgents and Phantom Zone prisoners.
The only indication that Zod may have aged more than his other crew members is the addition of greying facial hair that Zod did not have at the beginning of the film. Without facial hair at the beginning of the film, there is no way to know that the greying would not have been present had he had facial hair then as well. Apart from the goatee, Zod show no other indications of age. No additional wrinkles, no greying hairs on his temple, sideburns, or speckled throughout his hair.
Zod's age can be accommodated by explaining that Kryptonians have mastered their own genes to maximize their lifespan. Alternatively, as their ship includes hibernation and stasis, the crew may have taken shifts within stasis while they wandered the universe.
If we accept the premise of the question and this latter explanation, then Zod's age differential would be readily explained by his spending less time in hibernation than the crew in order to oversee and run things. As proven by the Earth incursion and his attempt at a coup, Zod is hands-on and leads from the frontline. He did not sit in his command and have his underlings do his bidding, but instead entrusted the key aspects of his plan primarily to himself. With that mentality and characterization, it would be difficult for Zod to cede command while in hibernation too often, and thus he would not get the benefit of stasis sleep.
Followers change. They could be newer, younger soldiers. The burden of responsibility can also grey a person, just look at at U.S.A presidents before and after their term.