I am forgetting the movie already, so before I answered to the question, when would it be a good time to do a search, I have a question - when did Khan reveal his name?
About 45 seconds before Sulu picked up USS Vengeance warping towards them on their sensors. By which time, Kirk is a lot less interested in who Khan is than he is in what the hell is going on between Khan and Marcus.
Actually, even without looking up Khan's background, both Kirk and Spock come to relatively clear conclusions about who he really is. During the gun battle to take the Vengeance, for example, Kirk orders Scotty to "drop him" as soon as they take the bridge. Scotty says "I thought he was helping us?" to which Kirk replies "No, Scotty, I'm pretty sure we're helping him."
Oldspock didn't tell NuSpock anything he didn't already know, except that Khan's singular weakness is his single-mindedness: once he's dedicated himself to a particular objective, he'll usually follow it even if that objective rolls off a cliff.