Au dernier rang s'eleva la voix de Markos Ratatopos, un élève grec arrivé depuis peu au collège.
Why isn't there any relative pronoun before "arrivé"? As far as I know relative pronouns should never be dropped in French as it's the case sometimes in English. I also wonder why the helping verb (etre) is omitted also. I thought about "arrivé" being an adj. but after checking several dictionaries, I couldn't find it as an adj.
Your last thought about it being an adjective is the closest to the mark.
Most past participles can be used as adjectives of their "patient", to use the term somewhat loosely. The patient is usually the direct object of a transitive verb, but for Dr. Mrs. Vandertramp verbs like arriver the subject fills this role.
Le ciel vu à travers la fenêtre était nuageux.
La femme née à Paris avait déménagé en Angleterre.
Once you see what's going on, I think you'll actually find it easy to read since it mirrors English:
The sky seen through the window was cloudy.
The woman born in Paris had moved to England.
Do note that what is strictly called the passive ("The book was written by the author") is not as idiomatic in French as it is in English, and the same goes for the elliptical version. If anything, the Dr. Mrs. Vandertramp verbs like the one you quoted might do this more naturally, though I invite native speakers to comment.
