He attended Mount St. Michael Academy before entering Iona College, but left in his sophomore year to study for the priesthood at St. Joseph's Seminary in Yonkers. He earned a Bachelor's degree and a Master's in Divinity from St. Joseph's.
Bishop Sullivan was ordained by Terence Cardinal Cooke on May 29, 1971, and then sent to the Dominican Republic to learn Spanish, so as to better serve the Spanish-speaking Catholics of the Archdiocese of New York. He spent three months in the Dominican Republic, where he "learned to speak Spanish very fast because nobody spoke English."
Upon his return to the United States, Sullivan served as parochial vicar at St. Elizabeth's Parish in Washington Heights for five years. He also served as parochial vicar at Ss. Philip and James Parish in the Bronx (1976–1981) and at Ascension Parish in New York City (1981–1982). He was named pastor of St. Teresa's Parish on the Lower East Side in 1981, where he gained a degree of fluency in Chinese In 1999, he was raised to the rank of Honorary Prelate of His Holiness. Then, in 2004, he became pastor of Ss. John and Paul Parish in Larchmont.
Bishop Sullivan was also a member of the Lower East Side Catholic Area Conference, Archdiocesan Priests' Council, and Archdiocesan Review Board for sexual abuse cases.