Kundukulam completed his ecclesiastical studies at St Mary’s Major Seminary in Trichur and at St. Joseph's Pontifical Seminary in Aluva. Kundukulam was ordained priest on 20 December 1942. Kundukulam was a member of the pastoral ministry until he was nominated to become a Bishop. He also served as the director of St. Anne's Charitable Institute from 1958 to 1970. In 1967 Kundukulam founded St. Christina's Home in Pullazhy, which a rehabilitation center for Unwedded mothers.
Kundukulam had a close relationship with Congress Leader K. Karunakaran. During the visit of Pope John Paul II to India in 1986, a visit to Thrissur had not been scheduled. Mar Joseph Kundukulam sent a request to Karunakaran to bring the Pope to Thrissur and Karunakaran was able to schedule a papal visit to Thrissur. Vinod Rai was the District Collector of Thrissur that time, and the necessary arrangements were made for the papal visit. Pope visited Thrissur on February 7, 1986.