If you're going to "put on the dog", that means you plan to get seriously dressed up.
The earliest reference in print is from a book written in 1871, entitled Four Years at Yale, and indicates that the phrase may have begun as college slang.
It explains, “To put on dog is to make a flashy display, to cut a swell.”
|Manet's Portrait of a Man |
(public domain, Wikimedia Commons)
We wonder if that's why the stiff stand-up shirt collars that were the height of male fashion in the 1870s and 80s were known as "dog collars"!
Sources: History.org, Online Etymology Dictionary