It was an explosion of emotion from somebody who, quite literally, lost everything important in his life. Go through half of what he has done and see how you turn out.
They're Vulcans, and even though Spock is half Vulcan, that outwardly cold exterior rarely thaws. Emotionally and with respective to bonds (because I'm not sure how Vulcans feel love or attachment to others) there are three people who are the most important to Spock: Sarek, Spock's mother and Uhura.
If Uhura had died at the hands of Khan, I could buy into Spock's meltdown because I have the feeling Spock relies on Uhura a lot more than he lets on. But the emotional connection between NuSpock and NuKirk just isn't strong enough or deep enough to warrant such a reaction. Just how long has been Kirk been captain? A few months, a year maybe?
At most we know that Kirk and Spock are colleagues who have a respect for one other, though for the most part both consider their counterpoint a thorn in their side. We need to remember that the Kirk and Spock in this universe have a friendship which is considerably less substantial than the one between Kirk and Spock in the prime universe.
So for Spock to emotionally crack like that, he would have to have a pretty deep bond with Kirk to elicit such a response. But if Spock considers Kirk a colleague whom he respects for the most part, that emotional connection ain't there. So did Spock crack because he saw another person, another crew member die? Was it one death to much? I'm just saying the friendship between Kirk and Spock seems to have been a little overstated and perhaps exaggerated on the part of critics and viewers alike.
Finally, it was admiral Marcus who placed the Enterprise in such a desperate situation with the warp drive not functioning, it was admiral Marcus who opened fire on the Enterprise crippling this ship. It was NOT Khan who inflicted such damage on the Enterprise, so why didn't Spock scream 'Marcusssss!!!' or something of that sort?
In fact I'm surprised by how many people have overlooked this!