There is no metaphor here. (Also, no chorus or bridge.) The whole song is a kvetch about schelpping "up and down the stairs at school."
There is no romance-- he doesn't pass someone on the stairs all day long and flirt with and/or get ignored by her while never getting the chance to actually converse because he is always rushing to class.
There is no bullying by being pushed down the stairs as he's climbing up, or class warfare (as in the British TV show Upstairs Downstairs)... or anything else.
Just a student weary of all the stair-climbing he is doing every school day. The repeated line is: "Up and down the stairs is driving me crazy!"
"Who thought education could be cruel?" he moans. "In the morning you'll find out that you'll/
Start out on the highest floor/ Then it's French in 104," presumably all the way down on the first floor.
"I would like to know who made the rule," he further bewails, "That each classroom, [from] door to door/ Is ten miles from the one before."
Like in the song "Wonderful World" (the Sam Cooke one, not the Louis Armstrong one), we also learn about the speaker's classes. Aside from French, he says, "I don't mind geometry/ English or biology" (one of Simon's weaker rhymes) and "I can wade through history/ Though it's just a mystery." So that's five classes' worth of stairs, plus lunch... and most likely, gym class.
As if he hasn't exercised enough for one day, poor dear.
That's really all there is to this cute little novelty number, with its nursery-rhyme score. You might expect it in the soundtrack of some movie set in a 1950's high school, like Grease. Although on-screen, there might be more happening-- on the staircases between classes-- than there is here.
Next Song: Charmed Life