Alberti was appointed as director (rettore) of Seminary in Cagliari in 1971.
In 1973 he was also appointed as secretary and professor at Pontificia Università Lateranense.
In 1987 he was elected to lead the church of Sardinia, being appointed Archbishop of Cagliari, becoming President of the Sardinian Episcopal Conference.
Alberti retired for age limit the 20 June 2003, and come back to his hometown Nuoro, where he died July 2012.
Alberti was an historian who wrote several books, mainly about the history of Christianity and the Roman Catholic Church in Sardinia.