Frank Sinatra was termed by some a functioning alcoholic:
Frank Sinatra was a functioning alcoholic, who went cold turkey on his
love of booze and cigarettes in the weeks running up to recording
sessions to save his voice...We gathered information that, from the mid-40s on, he was really seriously abusing the booze, and, even in his later life, drinking a bottle of Jack Daniels a day.
The daily beast elaborates in the article How Frank Sinatra drank American Whiskey His Way:
While Sinatra famously liked to enjoy a glass of whiskey (especially
on stage)...Long before he was hanging out in Vegas and Palm
Springs with Dean Martin, Peter Lawford, Joey Bishop and Sammy Davis
Jr., Sinatra was quite comfortable inside a bar and learned at an
early age the allure of liquor. That’s because he literally grew up
inside a speakeasy...During Prohibition his father, Marty Sinatra, ran
a joint in their hometown of Hoboken, New Jersey...It was also in his
family’s bar that an eight-year-old Sinatra launched his career
singing for the patrons when he wasn’t shining their shoes. His
affinity for watering holes continued well after Repeal.
The phrase in your passage is referring to the fact that the character in question was sitting on their own at a bar drinking alcohol, a trait many would consider (rightly or wrongly) a sign of some kind of alcohol dependency. From your explanation of the relationship between the two characters it appears to have been said in jest, so is not necessarily pejorative.
The use of the word 'act' reinforces the playful use of the phrase, the woman indicating she doesn't believe this behaviour
comes naturally to the male character.