Kudrick was ordained a priest on May 3, 1975,[3] staying on at Saint Francis College as a member of the mathematics and computer science faculty.[2] In 1978, he was named director of the university's computer services.[2] He was director of postulants for the Order from 1976 until 1980.[2] Kudrick assisted at parishes of the Byzantine Catholic Archeparchy of Pittsburgh during his time as a Franciscan friar, and eventually petitioned for incardination into the archeparchy, which he received in May 1987.[2][3][4] He served at several parishes as a priest under Archbishop Judson Procyk, and his last assignment was as proto-presbyter of the Cathedral of St. John in Munhall from 1998 until 2002.[2]
When Procyk died on April 13, 2001, Kudrick was elected administrator of the vacant Archeparchy of Pittsburgh on April 24, 2001. He served as administrator of the Archeparchy of Pittsburgh until the installation of the new Archeparch, Metropolitan Basil Schott, on July 9, 2002.[2] Kudrick was consecrated a Bishop by him a day later on July 10, 2002.[3]