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I don't recall if there was anything in the movie that established it. However, assuming that the construction of the Enterprise itself was not significantly changed due to the loss of the Kelvin then its maiden voyage happened in this film (Pike says as much when they first leave Earth). And so this is the final year (one presumes) of Kirk's time at the academy.
In the original timeline when Kirk becomes captain the Enterprise had already been on one five year mission with Pike as the captain (and Spock as first officer). So that would seem to argue that the movie is happening at least 5-7 years before the beginning of TOS (that episode starts with the Kirk five-year mission already some amount of time under way).
And looking at the "canon" timeline I see that the events of "The Cage" happened in 2254 (this is the episode that used the original pilot with Pike as captain) while the first episode of TOS happened in 2265. So by that timeline, again assuming that the construction of the Enterprise was not moved, the events of the movie must be more like 10-13 years earlier than when Kirk became captain of the Enterprise in the originally presented timeline.
Stardates were mentioned frequently in the movie so it would be relatively easy to align with the events of the original timeline (unless destruction of the Kelvin triggered a change in what stardates mean).
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