Ellis Island (1892–1954) was a U.S. immigration inspection station in Upper New York Bay. Ships arrived there from Europe (including Italy); passengers were processed for entry.
It was not a ticket office or departure point for ships going to Italy. Transatlantic tickets were purchased in European ports (Genoa, Naples, Palermo, Trieste) from steamship lines (White Star, Cunard, Hamburg-Amerika, Navigazione Generale Italiana).
1913 context: Peak immigration year (1.2M arrivals). A steerage ticket from Naples to New York cost ~$30–$40 (≈£6–£8)—equivalent to ~$900–$1,200 today. But you bought it in Naples, not at Ellis Island.