Remember that intrō takes an accusative, and per also takes an accusative.
Since there are multiple accusatives next to each other, by default, I'd assign each one to the nearest "accusative-taker":
Is quī viā Latīnā venit [per portam Capēnam] [Rōmam intrat].
He who comes from the Via Latina [enters Rome] [through the Porta Capena].
And indeed, this default interpretation seems to make sense, since the porta Capēna is the name of an actual gate in the walls of Rome.
Another clue comes from the fact that prepositions tend to come before their nouns—hence the name—and verbs tend to come after. It's not a hard-and-fast law, especially in poetry, but it's a good rule of thumb to keep in mind.