The only time the speaker really enters into the poem is in the first line, when he seems to read to us the "first of the undecoded messages." But he is best compared to the omniscient narrator of a story of novel. He knows everything that goes on, from the characters' thoughts to what Popeye is doing out in the country. He likes to tell riddles that can't be solved (or, if you want to get technical, Chinese tangrams that can't be put together), which is a like telling a joke without a punchline.
He uses all the usual words to tell a story, like "Meanwhile" and "Suddenly," but there isn't much of a story to tell. People come into the apartment and leave. There is lightning outside. And so on. For the most part, he stays out of the way of his characters, who use much fancier, more highfalutin language than he does. If you compare the parts of the poem outside quotation marks with the dialogue of the characters, you'll see that the characters are way more pretentious-sounding. Sentences that begin "Henceforth shall Popeye's apartment be but remembered space" can't be blamed on our narrator – they are just part of the story. Of course, if you want to blame the narrator for telling such a story that ends with someone scratching his balls, go right ahead.