This achievement can't be missed, as the puzzle is necessary for game completion.
*** Spoiler - click to reveal ***In the present, have Bernard pick up the "Help Wanted" sign in the first room of the Mansion. It's in the window to the left of the entry door.
You can get it right after the intro ends, and you get control of Bernard right at the start.
When you get to control Hoagie, switch to Bernard, and flush the sign to Hoagie in the past.
When Hoagie meets Red Edison in the cellar behind the clock-entrance, give him the Help Wanted sign to unlock the achievement (and get the lab coat).