A native of Naples, Papi studied at the conservatory in that city, holding various posts as conductor and choirmaster after his graduation. He came to the United States in 1913, becoming Arturo Toscanini's assistant on the conducting staff of the Metropolitan Opera in New York City. When Toscanini left the company, Papi became principal conductor, making his debut in that position on November 16, 1916 with Giacomo Puccini's Manon Lescaut. He left the Metropolitan in 1927 to become the first conductor of the Chicago Civic Opera, but returned to his old post in 1935.
Papi had collapsed and was found dead in his apartment just hours before he was to conduct a performance of La traviata at the Metropolitan on November 29, 1941; the evening, which was to be broadcast over the radio, marked the company debut of tenor Jan Peerce. Ettore Panizza instead conducted, and news of Papi's death was withheld from the cast until the performance ended.