Is a mother of a priest automatically given passage to heaven?
No. While a woman whose son becomes a faithful and devout priest should be greatly respected and who will certainly benefit from the prayers and sacrifices of her son, all human beings have free will and must freely choose to serve God to the best of their ability according to the light that they have.
In fact, in his book Honey from the Rock, a compilation of stories of Jewish converts, editor Roy Schoeman tells the story of Fr. Hermann Cohen, a Discalced Carmelite priest who deeply worried about the salvation of his mother, who apparently refused to believe in Christ and died without any perceptible conversion. His friend St. John Vianney urged him to have faith and, a number of years later, a local mystic told Fr. Cohen that she had received a vision that his mother had converted after unconsciousness and before final death at the intercession of the Blessed Virgin.