I went with the timeline from the Memory Beta article about the Garth of Izar novel for the name of the ship and the event at Antos which says early 2250s.
That doesn't make sense to me. "Whom Gods Destroy" refers to Garth as a "new inmate" on Elba II since the Enterprise's last visit there. His shapeshifting ability, which he learned around the same time he went mad, was unknown until he used it to escape confinement on Elba. I have a hard time believing that he would've let himself be confined for well over a decade before using the ability to escape and seek revenge. The implication is that his accident was a relatively recent event.
Not to mention that Garth was Kirk's hero when he was in the Academy, and Kirk graduated in 2255. It stands to reason that Garth's disgrace would've been well after that. If Kirk had spent his entire Starfleet career thinking of Garth as a former hero who'd already fallen from grace, then his reaction to Garth's fall wouldn't have been as surprised and sad as it was in the episode.
Then there's the fact that Steve Ihnat was only 34 years old -- younger than Shatner -- when he played Garth. Granted, he did refer to "you Earth people," so he may have been intended to be a humanoid alien (though the novels have made Izar an Earth colony), which could give him greater longevity. But there's no proof of that, and Trek humanoids are generally treated as having humanlike lifespans unless it's a specific plot point that they live longer. So the casting suggests that Garth was meant to be not too much older than Kirk.