He didn't finish the academy in three years. He bragged that he would, jump cut to "three years later" and a few scenes in he's about to be kicked out for dishonesty. No evidence of him successfully completing a four-year program in three years.
The evidence is his battlefield promotion to captain of the Enterprise.
There's "hard worker who successfully completed four years of classes in three years" and there's "bad writing ass-pull". Kirk's was definitely the latter. Besides, if his promotion somehow means he actually did the coursework in three years, there's the problem of all this classmates who kept their commissions too. Did Bones and Uhura and Cupcake all also complete all the coursework in three years? It's Ockham's Razor.
I am not interested in an argument as to whether NuKirk or Kirk Prime is better, because for me, the obvious answer by far is the latter. While I like Star Trek 2009, I'm a TOS-Enterprise era Star Trek fan. And STID was horrid to me. SO, I am not a Abramsverse apologist.
But that all said, you asked for evidence NuKirk completed his academy training. He did. He completed the curriculum in 3 years, not 4, and while his status was in question due to the Kobayashi Maru incident, no formal punishment was handed down because the hearing was interrupted. So, you can't claim he was kicked out of Starfleet.
What we do know is that Pike DID promote him to First Officer, during wartime. Kirk later ascended to acting captain after relieving Spock of command, and by the end of the movie, the promotion was made permanent. Both promotions, and the heroism supersede the pending Kobayahsi Maru incident, which we don't necessarily know would have prevented him from graduating, anyway. In the prime universe, Kirk got a commendation for "original thinking," and I think he likely would have in the Abramsverse, as well.
Just because Spock was pushing for Kirk to be reprimanded doesn't necessarily mean he would have been. In fact, Kirk's point of view was that it was a cadet's job to prove a test that is claimed unbeatable, was beatable. No one said Kirk couldn't reporgram the test, and the fact he was capable of reprogramming a test designed by Spock actually proves Kirk can out think Spock.
You can call Kirk's promotion an ass pull, bad writing, or ex Deus Machina, and you'd get no argument from me. But as horrid as STID was, it DID prove Kirk wasn't quite ready for the captaincy, though he redeemed himself by the end of STID. Bby the same token, with Kirk in command, Earth literally was saved arguably twice (Beating Nero, and later beating Khan and Marcus and preventing a possible war between the Klingons and the Feds...it wasn't clear the Feds could beat the Klingons in a long term war, and almost every alternate timeline shows the Klingons can and do beat the Feds).
SO the bottom line is that Starfleet felt saving Earth was the ultimate in demonstrating Kirk had learned what he needed to know to be a Starfleet officer, and his promotion to captain was his graduation.