This achievement can only be obtained after you finish the main story.
In the final Special Assignment "Fly Me To The Moon", you can obtain the Rex Fury character token. This is done after you have beaten the final boss, which is also Rex Fury. The final part of the mission is a free fall portion as you are falling back toward LEGO City. Toward the end, you will fall through a rocket with energy beams. Once this part is over, quickly move over to the right side and make sure to go through the first stud circle that appears. In the middle of this circle is the Rex Fury character token. Once collected, either finish the mission or save and quit back to LEGO City.
Now back in LEGO City, go to a Disguise Booth in any district or to the basement in the police station. Here, switch out your robber costume to Rex Fury.
Now to find a lawnmower. I found mine in Crescent Park, so I suggest heading over to Crescent Park, which is the light green district on west coast of the island on the right. Simply, hit
to hop onto it when in the Rex Fury disguise and the achievement should pop within a few seconds.