Prime Khan did read the Enterprise engineering specs (I hope it was just the version for public reading) on his own in "Space Seed" and became familiar with the ship's layout and functions.
Ship systems might have changed enough in fifteen years that he couldn't immediately find the override for an obscure remote shield function after being in the captain's seat for a brief time.
Presumably, NuKhan was receiving much more focused and specialized technical training during his months with Admiral Marcus and Section 31. Under those circumstances, he could probably become very proficient at Starfleet technology.
Kor
Which also begs the question of why did Marcus go through all that trouble with Khan. Yeah he did have a supposed superior brain but with all the species Starfleet had encountered couldn't marcus have more easily found someone that was a ruthless genius that grew up with the modern technology to create him some new WMDs. Why not just steal some alien advanced tech? It seems to thaw Khan out, retrain him by threatening to kill him people and then ordering him to design new stuff would take a long time. Even though both incarnations of Khan are geniuses Prime Kirk beat Prime Khans butt twice and proved himself smarter than Prime Khan. And we know in the NuTrek timeline that NuKirk is a genius. What I am getting at is that Marcus should have been able to find a genius somewhere that would have been willing to help him. He had a whole ship of evil Starfleet officers willing to help him on the vengeance. I am sure he could have found a somewhat morally corrupt Starfleet officer that was the equal genius of NuKirk.
Trying to find someone to go up specifically against nuKirk seems rather myopic, considering that when Marcus released Khan, that was months before Kirk essentially volunteered for the mission. Kirk wasn't Marcus' primary enemy or objective, he was just a pawn that fought back.
But Marcus was vindicated in picking Khan over a familiar alien or the tech available to them (Starfleeet, after all, is supposed to be a composite of all sorts of tech -- ENT established that it's not all human tech, but rather tech boosts from various member worlds, exploration, and encounters). With rather mundane tech like a rifle and a ship-mounted energy weapon, Khan was able to destroy several Klingon shuttles and defeat an entire platoon of Klingons by himself on foot -- and in canon by this point in history, the Klingons were deemed to be more powerful and much more of a threat than the Xindi (a neat twist by Enterprise, I felt). Khan being able to do that with garbage is highly impressive, but for all we know, Marcus *did* give Khan advanced alien tech to design the Vengeance. Khan, after all, had access to the transwarp beaming device, which only a select few had access to and was one of the top technologies at Starfleet at the time.
Also, no alien race in Prime Trek came up with ship-mounted arcing energy weapons before, not even the Borg; there are plenty of examples of magnets or collectors for certain types of energy, but not really weaponized like that. They *could* be a Khan invention, which is also impressive and the evidence of the kind of unique but blood thirsty thinking that Marcus was looking for. Quality of knowledge is one thing, but application is another.