Google suggests "Old Duke wine" when you search for it and comes up with this image: https://www.etsy.com/listing/210039515/old-duke-red-grape-wine-growler There's also this forum post if you go digging: "Old Duke was the nastiest wine I ever drank. As I remember it, there was a drawing of a drunk leaning up against a street lamp as the picture on the label. If it's not made anymore, be REALLY thankful!" (https://answers.yahoo.com/question/index?qid=1006011303497 I'm going with "cheap alcohol."

I don't know either. I'll vote for - the name of a cheap, old-fashioned after-shave lotion!

I am going to take a wild guess; I've never heard the expression; I'm wondering if it might be rhyming slang for "puke."

Just a guess but is it some sort of cheap alcohol?