I don't ever remember Chekov in the original series ever have anything to do with engineering. Is there an in-universe reason for him being there?
Chekov wasn't in the first series (of the original series) when the Enterprise encountered Khan and, so Khan shouldn't have recognised him in The Wrath of Khan. This is a well known meme among fans.
The idea behind putting Chekov in engineering was to provide an explanation as to why Khan may have recognised him. He was on the Enterprise, but was a junior ensign in engineering and not on the bridge, so Khan could well have seen him.
In short, it's one of those nods to fans of the original Trek that appear in the reboot (like comments about Admiral Archer's prize beagles).
It is standard policy in most naval services for personnel to be cross trained in a variety of ship services. For example, if you read through the Wikipedia entry for the Submarine Warfare Insignia there is this comment about the US Navy version:
To earn the right to wear the pin, prospective submariners complete an extensive qualification process that lasts about one year (for both enlisted and officers, though the two programs differ significantly) and covers virtually all of the submarine's systems.
Where "virtually all systems" is the key phrase.
The reason for this is that, while deployed on a mission, the people the ship has are the only ones that it's going to have. It may go weeks, or even months without seeing a port where it can replace personnel.
Kirk asks him about his shadowing of Mr. Scott; it's clear he has been cross-training into engineering.
For navies, cross training is fairly common, at least to apprentice levels (Keeping in mind that apprentice is the lowest rating actually holdable). Many people in the USN are rated in 2-3 fields, albeit their best rating is their formally trained field; a significant minority take free-time to train in a more desirable field than the one they were trained in, and "strike" for a new rating in that field; not a few actually get reassigned into the alternate rating.
Further, on board ship, the potential for promotion by attrition is much better if you can fill multiple different people's jobs.
Further, Chekov shows some clear engineering talents in ST XI - his ability to do things that even a much more experienced specialist can't pull off - with the beaming of the falling targets.
So, Chekov has good career reasons to cross-train - promotability and survivability - and has been shown to have natural talent in engineering. Cross-training is a normal part of the inherited naval traditions. And it's confirmed by dialogue that he has been cross training. Kirk further knows that Chekov is second only to Scotty in raw natural talent.
Therefore, it's reasonable that Kirk would put him in charge of Engineering.