This "mono" doesn't seem related with "to go cold turkey", whose correspondent expression is rather made with "quitarse" or "tener".
Here, "ponerse el mono" rather seems to be talking about the piece of clothing, like "to put on the overalls"... that would mean to have an attitude that shows disposition for working.
Btw, "ponerse mono/mona" would be to be to get pretty, to become very kind, to show oneself nicer than one usually is.