The speaker opens by calling themself "old hat," meaning outdated, familiar, nothing new. But the poem quickly plays with that phrase. A hat can be a literal thing, and the speaker seems to picture themself as one: a "quaint nest" where "the lines meet." Those lines could be the lines of a poem, or the creases and seams of a well-worn hat, or even the lines on an aging face. The poem doesn't say, and the ambiguity is part of the fun.
The tone is light and a little cheeky. Instead of sounding embarrassed about being old-fashioned, the speaker says "look at me go," as if showing off. The last lines, about sandwiching "this ear with that," suggest a hat pulled snugly over the head, or perhaps a mind where two ideas are squeezed together. The mood is warm, humorous, and self-accepting. Being old hat becomes something comfortable, even a bit proud.