I can't remember, but I thought they had in the film. If so-wouldn't he then learn everything from Spock regarding the former timeline?
Kirk and old Spock do Mind Meld in the Star Trek 2009 movie.
Here is a link to the scene. In the scene old Spock shows Kirk how he ended up time traveling with Nero to the past.
In the movie its pretty clear the sharing went 1 way, Old Spock to Kirk, and that Old Spock also limits what he shares to Kirk to what we the viewer sees.