Well, Robert Service is a pretty straightforward guy, and here he basically just tells us what this poem is about: the end of a year. Still, that's important. The title sets our expectations for our reading experience. Before we even dive in, we know, more or less, what we're going to be dealing with.
The specific words in the title matter, too. He could have called this poem "New Year's Eve," or something, but that's not really what Service focuses on. He doesn't focus on the new year, but the old one, which leads to a slightly more serious and melancholy tone. And this poem is about what the "passing" feels like. Service freezes that one moment, and forces us to look really hard at the feelings that go along with it.