[tojoson]

There had to be a long period after they graduated before Kirk became a Starship commander. He didn't come right out of the academy as a captain.
There had to be something like 20 years or more between times, so he could have been serving on another ship while Spock was serving on the Enterprise with Christopher Pike.

Also, don't forget, Amanda (Spock's mother) was Human, not long lived Vulcan. Jane Wyatt was mid-fifties when the original Star Trek was made, so that would have made her mid thirties, or younger, when Kirk and Spock were at the academy, which fits well with Winona Ryder's age.