Spock actually failed that one. As first officer, the job is to delegate and organize. Spock should *not* have been in that reactor. He should have been ordering an engineer (who is less vital to the ship) to do it instead.
Also, he wasn't even first officer during WoK. At that point Spock was captain of the Enterprise, which makes it doubly inappropriate.
FTFY
No Kirk was Captain of the Enterprise at that point in TWOK, Spock was only Cpatain at the start of the film we have this line
An emergency situation has arisen. By order of Starfleet Command, as of now, eighteen hundred hours, I am assuming command of this vessel. Duty officer so note in the ship's log. Plot a new course, for Space Laboratory Regula I. Engine room. Mister Scott!