This achievement is missable!
Towards the end of the DLC the path you must follow to progress the story will bring you into sight of Rusty Rat for the second time, the first time you saw him getting chased by Carbon Cat. This time he will be tied to a rocket. After viewing him you must not die or take too long or you will fail the achievement.
To save Rusty Rat follow the main path, in the first room you enter (there will be a drone) head counter clockwise and take the lift opposite of where you entered. The lift will take you to Rusty Rat, interact with him and the achievement is your.