No.
We know Kirk takes the Kobayashi Maru test, so he is at least a Lt. Also, he is wearing an officer's undershirt, not a cadet uniform.
That's a test given to cadets at the academy.
Based on the statements given in interviews about the movie that Kirk was a cadet and in trouble at the academy, which is why McCoy had to help him out to get him on the ship.
There's no reason to assume he can't be a Lt. at this point.
Based on what?
From TWOK, it's possible that only people who have achieved at least that rank are even allowed to take the test.
Saavik was wearing cadet red and whenever the ship and its mission were mentioned, it was referred to as a ship full of cadets and trainees on a training mission. And since in this is a new universe, we can't really know for sure at the moment, can we? Especially since there are no references to Kirk being an officer that I'm aware of at least.
Even high ranking officers have to go to school again in order to get promotions. My guess is that he's a mid-level officer on some sort of probation. I'm pretty sure the KOM test is a pretty advanced command test, not the sort of thing they let ensigns or raw cadets with no field experience take.
That'd be exactly the kind of test to give cadets just about ready to make the transition from cadet to officer.
I assume the writers of this movie are at least half as smart as me

, so if I can come up with several scenarios that make it work ok, they can too. We won't be disappointed, I think.
Again, and explanation they think up at this point would be contrived because of how they've set things up.
I don't see any Lt. Commanders or higher on the bridge. It's entirely possible that there aren't any fit for duty who also have the proper training. I'm sure if there is a Lt. Commander down in a biology lab somewhere, they would never let him near the bridge before a Lt. with advanced command training.
All officers have at least some command experience. But that isn't the point, the point is that there should already be an established chain of command and that Kirk is not qualified to be in it.
Except that he receives a field commission from Pike when he's promoted to XO after the first battle with the Narada.
Except Kirk isn't qualified for that position.
Pike likes him and respects his ability.
And knew his father and probably knew him growing up, but none of those are legitimate reasons to give a cadet a field promotion to XO of the ship.
Kirk has the same training as all the others, he just lacks the rubber stamp of a full commission.
We don't know that, but either way that would still make him the most junior officer on the ship because of a little thing called "time in service". If one ensign commissioned a month before another one, the former would outrank the latter. That's part of why having T'Pol act as the NX-01's XO didn't make any sense either.
Pike gives him that based on merit and lack of other options.
I'm not really seeing that. There's a lack of information either way, but there are certain facts that are known, like a cadet not being qualified to command a ship and field commissions can't be handed out on a whim, not in a responsible uniformed service.
After that he's an officer, second in command after Spock, and has every right to take command.
He's a cadet and not qualified to do anything more than lend a helping hand at a station on the bridge or another post elsewhere in the ship. He's a warm body, that's it.