There is a beautiful tradition for this day, at 1200 noon. Once upon a time one could obtain this day a plenary indulgence by reciting the Supplication to the Madonna of Pompeii. The other day for this is the first Sunday of October.
With the changes to the concessions for indulgences, according to the Enchiridion Indulgentiarum, there is no longer any plenary indulgence for this prayer, notwithstanding anything you might see in some old book or on a website. For example, if you see something about Pope Leo XIII granting an indulgence, etc., that is null and void now.
However, the new Enchiridion says with concession #17, §3 that Marian prayers obtain a partial indulgence under the condition that the prayer is approved by competent authority and that it is recited with fervor in the state of grace (you don’t need confession and Communion within 8 days, nor must you recite the prayers for the Roman Pontiffs intentions for a partial indulgence). You can receive a partial indulgence, by maintaining this beautiful custom of the Supplication today.
I called the office of the Apostolic Penitentiary in May 2007 to confirm this.
The text of the prayer to recite is below (in English). The language may seem florid to 21st century ears and tongues but it is vigorous, serious, and super Catholic. We need more unabashedly fervent prayers like this, friends. If you have a hard time reciting this, you can listen. I attached an audio file of the prayer to this entry.
Brief background: This devotion was started by Bl. Bartolo Longo, who had once been a Satanist "priest". He converted, did penance, and became a lay Dominican. In 1872, Longo, a lawyer, went to Pompeii see to the affairs of Countess Marianna Farnararo De Fusco. He started there a Confraternity of the Rosary. They obtained a picture of the Blessed Virgin from a monastery, which before that was in a second hand store, before which they could recite the Rosary every day. It showed Mary with Sts. Dominic and Catherine of Siena. In 1875 Longo received permission to build a church. Miracles were reported and pilgrimages began. The picture has been restored several times since then. Longo also saw tot he building of complex for works of mercy with orphans and prisoners. Pope John Paul II beatified Bartolo Longo in 1980. Some of his writings form the basis of the Luminous Mysteries of the Rosary.
You reach the sanctuary easily by walking just a few minutes out the back gate of the ancient ruins of Pompeii, famously destroyed by the eruption of Vesuvius in AD 79.
Here is the prayer to be recited (twice during the year). It obtains, under the usual conditions, a partial indulgence.

Supplication to the Madonna of Pompeii, 8 May and 1st Sunday of October [8:46m]:
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Here is the indulgenced text (written by Bl. Bartolo Longo), to recite at noon:
PETITION TO OUR LADY OF THE ROSARY OF POMPEII
In the name of the Father and of the Son and of the Holy Spirit. Amen.
I.
O august Queen of victories, Virgin who reignest in paradise, whose mighty name causes heaven to rejoice and hell to tremble, o glorious Queen of the most holy Rosary, we, thy happy children chosen by thy goodness in this century to build thee a temple at Pompeii, kneeling at thy feet on this solemn