"'By the sacred anointing of the sick and the prayer of the priests the whole Church commends those who are ill to the suffering and glorified Lord, that he may raise them up and save them. And indeed she exhorts them to contribute to the good of the People of God by freely uniting themselves to the Passion and death of Christ.'"
Catechism of the Catholic Church, n. 1499
Many people use Anointing of the Sick and Last Rites interchangeably. However, they are not interchangeable. The Sacrament of Anointing of the Sick is for those who are seriously ill (they need not be dying) as soon as they become seriously ill. The person should contact Fr. Anthony to arrange a time and place for the celebration of this sacrament (Fr. Anthony can go to one's house or the hospital room, in addition to the celebration of the sacrament at the parish).
Last Rites are the sacraments of Penance and Anointing of the Sick, as well as Viaticum (the Eucharist for the dying). This is done, as much as possible, when a person is dying and will no longer recuperate, but does not need to be at the last hours of life (there are separate prayers for that called the Prayers for the Dying). As soon as the person begins the pilgrimage towards death and there is no cure for the illness, the person him or herself or the family should contact Fr. Anthony to arrange for the celebration of the Last Rites.
When the person is within the last hours of life, Fr. Anthony can also come again and pray the Prayers for the Dying.