I'm in my 30s and I grew up in Sydney. I have never ever heard the phrase "fair suck of..." until your post. Mind you, the first time I had ever heard any phrase involving sauce bottles was Rudd's famous usage. Which I feel brought it back into common parlance. With people mostly using it as an ironic exaggeration of country slang, and in tribute to Rudd.

It's worth noting, that Rudd was an actual country kid. While certainly not representative the average Australian, he would definitely have been exposed to more authentic bush slang than most.