"The Passing of the Year" starts out with the speaker enjoying a quiet, relaxing New Year's Eve. He's at home with his pipe and a glass of something tasty. It's a rare moment of calm, and he's taking the time to look back on the year that's just about to be over.
As he thinks it over, he starts to weave an elaborate metaphor. He imagines the "Old Year" as an old, tired actor on a stage who's about to take a final bow. From there, he goes even deeper, as he pictures himself in the audience at this imaginary play. He looks at the other people around him. As they look up at the "Old Year," some of them are crying, some are smiling. The speaker guesses at their stories, and imagines all the things that have happened in this past year.
Then, the speaker takes us back to his house. His drink is done and his pipe is out, and it's time to go. He ends the poem with a friendly goodbye to the old year.