This achievement is gained during the course of the main story quests, about 25 minutes - 1 hour into the game. As far as I know, it cannot be missed.
Spoilers:
*** Spoiler - click to reveal ***After escaping the mine, you eventually will find an area that is gated off, guarded by several sentries and a merchant. Attempting to talk with everyone should eventually prompt you to go into the nearby Mine entrance (not the locked door, and not the one you originally entered outside from, at the top of the hill)
Inside this Mine section you will find an NPC named Merre. If he doesn't talk to you, try going outside and talking to all of the NPCs and leaving through the gate.
--Talking to NPC "Porre" seems to trigger this quest being available but I am unsure at time of writing. Eventually someone/something should prompt you to speak to Merre.
-Inside the mine, Merre will tell you about a rat problem.
-You will then go up the nearby pathway, break the nearby rotten boards to gain access
-Inside you will be in a room where rats spawn infinitely. You can turn the difficulty down to easy here if you are having trouble. You should defend yourself against the rats, but don't go out of your way because they will spawn infinitely.
-The only way to stop them is to press and hold the Z key (default keybind) to drag nearby Stone Bricks (on the ground) into the rat holes (which have some straw and other debris near their entrances, you should be able to spot them if you look)
--They should automatically "snap" into place when you drag them close enough.
--After plugging all of the holes, you should be instructed to speak to Merre again, and the achievement should unlock.
(You can also refer to the wiki if you need extra help:
https://nehrim.fandom.com/wiki/Work_in_the_Mine)