and for Mike I just stroll into the PARK LANE Liquor Store and ask for a bottle of Strega and
Patsy got a gift, why shouldn't Mike get one, too? Patsy gets a book, Mike gets a bottle of Italian liquor called "Strega."
In contrast to the bookstore quandary, the speaker knows right from the start that he's going to buy Mike alcohol. He doesn't have to put himself to sleep debating whether to get Jack Daniels or Grey Goose. He "strolls" into the Park Lane liquor store, buys the booze, and leaves. Easy.
Who are Patsy and Mike? Do they know each other? We'd guess they are a couple, maybe the couple that the speaker is going to see in East Hampton.
then I go back where I came from to 6th Avenue and the tobacconist in the Ziegfeld Theatre and casually ask for a carton of Gauloises and a carton of Picayunes, and a NEW YORK POST with her face on it
Reading this poem is like watching a game of PAC-MAN. The speaker is popping in and out of stores, picking up and putting down books, and retracing his steps.
He walks back where he came from and goes into a tobacco shop inside a Broadway theater called the Ziegfeld. He "casually" buys not one, but two different kinds of cigarettes, as well as a New York Post newspaper.
Oh, and by the way, the newspaper has "her face on it."
Whose face? "Lady's." That is, Billie Holiday's. The newspaper, evidently, is reporting the news of her death.
This is the moment when the speaker learns that, in fact, Lady Day has died, and his reaction is…nothing. Zilch.
In fact, the speaker does everything he can to sound cool and nonchalant, down to "casually" asking for the cigarettes. He slips her death into the poem like a minor detail, just another part of the day. This is ironic, because it's clearly not just another day. One of his artistic heroes has just died.
Can we just note that the speaker must be a big smoker – he buys cigarettes by the carton and not by the pack. He also buys Gauloises, an extremely potent cigarette brand associated with French intellectuals like Jean-Paul Sartre, who made them into one of his trademarks.
Finally, why isn't the name of the theater capitalized, like all the other places in the poem? Curious.