According to "The Making Of Star Trek" (Whifield 1969) and other sources, stardates were intended to
avoid using dates and years and to sound "futuristic," but certainly became confusing. Sometimes they'll talk as tho the 20th Century was "two centuries ago" and some times "300 yrs ago". They certainly weren't intended to hold up to fannish scrutiny! And the tech in "The Cage" was still very unfixed in terms of the ST that came later, borrowing more from "Forbidden Planet" than anything else. It was only later that the stardates really became linear, especially TNG onward.
It's a bit of a fudge, but one could imagine warp technology evolving over the years, and in Pike's time there had been a breakthrough to Warp 7 or 8 (like the Warp 5 barrier ENT talked about) that made the ride different until the tech got better. Remember, in the full pilot (not aired as part of "The Menagerie," so therefore not "canon"), when Pike orders the time warp the bridge gets translucent and Kelso signals with his fingers that they hit warp 7, as if they couldn't speak.
I'm sure that later in life Kirk would have quoted the esteemed Dr. Henry Jones Jr. - "It's not the years, but the mileage." All that warp travel, plus occasional time-travel, not to mention alien posession, body transfers, mugato poisoining, etc., must have aged a person. I'm sure it wasn't intentional, but Kirk's age is never mentioned in the movie era. After all that travel, who knows.
Just found this interesting page on Stardates:
http://startrekfactcheck.blogspot.ca/2013/07/captains-log-stardate-unknown.html