The full explanation of the Rosa Mystica devotion is on this website at CatholicTradition.org.

The devotion does not seem to specify exactly which prayers are said, so below is my suggestion. You can substitute whatever is, in your good judgment, the best prayers, sacrifices, and penances to do on these days.

I suggest, daily, the following 7 prayers, in this order:

Golden Arrow Prayer
May the most holy, most sacred,
most adorable, most mysterious,
and unutterable Name of God
be always praised, blessed,
loved, adored and glorified,
in heaven, on earth and under the earth,
by all the creatures of God,
and by the Sacred Heart of Our Lord Jesus Christ
in the most holy Sacrament of the altar.

True Holy Eucharist Prayer
May the most Sacred Heart of Jesus
be adored and loved
in the true Holy Eucharist
at every Mass
throughout the world
until the end of time.

May the most Sacred Heart of Jesus
be adored and loved
in the true Holy Eucharist
in every tabernacle
throughout the world
until the end of time.

May the most Sacred Heart of Jesus
be adored and loved
in the true Holy Eucharist
by the heart of every Christian
throughout the world
until the end of time.

Rosary
[at least one decade]

Rosa Mystica Prayer
Blessed Virgin Mary,
Most Pure Rose of the Most Sacred Mysteries,
obtain the Mercy of God for us
through the Sacred Heart of Jesus
and grant us repentance, conversion, and healing
through your Motherly Immaculate Heart.
Amen.

Divine Mercy Chaplet
[at least one decade]

Prayer of the Publican:
O God, be merciful to me, a sinner.
[O God, be merciful to us, poor sinners.]

Seven Words Prayer
Jesus, Mary, I love you, Save souls

.

Pray the above 7 prayers daily for the first 12 days of each month. Please keep in mind the intention of the grace of chastity and purity for clerics, religious, and the laity — married as well as single.

On the 13th day of the month, pray extemporaneously (speak to God and Mary from your heart, in whatever words you may choose) and ask God for whatever favor you may desire. Celebrate the 13th day as a day dedicated to the Blessed Virgin Mary.

Mary: “I wish the 13th of each month to be celebrated as the day of Mary. I wish the 13th of each month to be celebrated as the day of Mary. On the twelve preceding days, special prayers of preparation should be said.”

Remember to pray also for the intentions of the Roman Pontiff. If you disdain or reject the Roman Pontiff, whosoever he may be, may God punish you. Those who have true faith believe that each and every Roman Pontiff will never err gravely, never fail in faith, and never lead us astray.

[This particular form of the Rosa Mystica devotion
and the Rosa Mystica Prayer itself was composed by Ronald L. Conte Jr.]