In the first movie, there were clear indications that more time passed than it appeared, yes. But here,
Anyways, the video game theoretically happens between Countdown to Darkness and Into Darkness - it starts with the Enterprise being diverted by a distress call (versus the normal mission start to Countdown to Darkness), and also ends with a trip to Nibiru. The video game covers 2259.32-2259.33 - if the Mudd comics ended the day beforehand, that would put it 24 days before Into Darkness, close enough to a month in my book.
That would account for it.
I wonder if there were late script changes giving it an earlier date and adding that scene with Pike, or whether this stuff wasn't actually being vetted as strictly as is implied.
I wouldn't be surprised if the comics writer is just given a list of details and plot points to tie into but isn't necessarily told what significance they'll have in the movie, given how secretive Abrams has been. And there may also be a list of things to avoid contradicting, but they may not think of every last detail. I know from experience how hard it can be to keep every last continuity detail straight between different works.
Yeah, I had that same thought myself. But it's a stretch, and I don't think it's what the movie's writers had in mind.