There is a short scene where Kirk ask Chekov about him following Scotty like a puppy for the last months.
There, mission accomplished.
The audience learned everything they need to know to accept Chekov in engineering.
Pretty much (in terms of explaining why Chekov could
do the job--whether he should
be doing it is a little less convincing, to me, but I chalk it up to the standard "must use each of the main cast in at least one 'moment' onscreen" rule that has been a part of cinema for, oh, over a century).
An argument could probably be made that the potential for Chekov to take over at least temporarily for Scotty was being set up even during the first movie. Not all of the mirroring/symmetry going on there involved Kirk and Spock.
True enough. But then, that would require people to carry over something from the previous film in order to better understand the new one. Can't be having that kind of nonsense.