I can think of two possibilities - neither which seems totally satisfactory...
1. There was a St. Bernard named Neal in a TV show from the early 50's called Topper. I guess he was actually a ghost-dog iirc, but he loved martinis. It was not a cartoon though...
2. The sheepdog from Tex Avery cartoons with the coyote. I don't remember him drinking per se, but he did used to clock out of his shift (of getting wiley coyote) and considering the era, perhaps had a cocktail or two...