What prayers to say?

Some of the typical prayers to be said during the holy hour are: If you posses a traditional missal, you will find plenty of prayers you could follow, including the Canon of the Mass. You could get a booklet with set prayers, like "Holy Hour of Reparation", Father Mateo's 40 hours devotion book, or others. Here is a list of places which have some, as well as other books about the devotion to the Sacred Heart:
Top Act of Reparation

O sweet Jesus, Whose overflowing charity for me is requited by so much forgetfulness, negligence and contempt, behold us prostrate before Your alter (in Your presence) eager to repair by a special act of homage the cruel indifference and injuries, to which Your loving Heart is everywhere subject.

Mindful alas! that we ourselves have had a share in such great indignities, which we now deplore from the depths of our hearts, we humbly ask Your pardon and declare our readiness to atone by voluntary expiation not only for our own personal offenses, but also for the sins of those, who, straying for from the path of salvation, refuse in their obstinate infidelity to follow You, their Shepherd and Leader, or, renouncing the vows of their baptism, have cast off the sweet yoke of Your Law. We are now resolved to expiate each and every deplorable outrage committed against You; we are determined to make amends for the manifold offenses against Christian modesty in unbecoming dress and behavior, for all the foul seductions laid to ensnare the feet of the innocent, for the frequent violations of Sundays and holidays, and the shocking blasphemies uttered against You and Your Saints. We wish also to make amends for the insults to which Your Vicar on earth and Your priest are subjected, for the profanation, by conscious neglect or terrible acts of sacrilege, of the very Sacrament of Your Divine Love; and lastly for the public crimes of nations who resist the rights and teaching authority of the Church which You have founded. Would, O divine Jesus, we were able to wash away such abominations with our blood. We now offer, in reparation for these violations of Your divine honor, the satisfaction You once made to Your eternal Father on the cross and which You continue to renews daily on our altars; we offer it in union with the acts of atonement of Your Virgin Mother and all the Saints and of the pious faithful on earth; and we sincerely promise to make recompense, as far as we can with the help of Your grace, for all neglect of Your great love and for the sins we and others have committed in the past. Henceforth we will live a life of unwavering faith, of purity of conduct, of perfect observance of the precepts of the gospel and especially that of charity. We promise to the best of our power to prevent other from offending You and to bring as many as possible to follow You.

O loving Jesus, through the intercession of the Blessed Virgin Mary, our model in reparation, deign to receive the voluntary offering we make of this act of expiation; and by the crowing gift of perseverance keep us faithful unto death in our duty and the allegiance we owe to You, so that we may one day come to that happy home, where You with the Father and the Holy Spirit lives and reigns, God, world without end. Amen.

Top Another Act of Reparation

Sacred Heart of Jesus, animated with a desire to repair the outrages unceasingly offered to Thee, we prostrate before Thy throne of mercy, and in the name of all mankind, pledge our love and fidelity to Thee!
The more Thy mysteries are blasphemed, the more firmly we shall believe them, O Sacred Heart of Jesus!
The more impiety endeavors to extinguish our hopes of immortality, the more we shall trust in Thy Heart, sole hope of mankind!
The more hearts resist Thy Divine attractions, the more we shall love Thee, O infinitely amiable heart of Jesus!
The more unbelief attacks Thy Divinity, the more humbly and profoundly we shall adore It, O Divine Heart of Jesus!
The more Thy holy laws are transgressed and ignored, the more we shall delight to observe them, O most holy Heart of Jesus!
The more Thy Sacraments are despised and abandoned, the more frequently we shall receive them with love and reverence, O most liberal Heart of Jesus!
The more the imitation of Thy virtues is neglected and forgotten, the more we shall endeavor to practice them, O Heart, model of every virtue!
The more the devil labors to destroy souls, the more we shall be inflamed with desire to save them, O Heart of Jesus, zealous Lover of souls!
The more sin and impurity destroy the image of God in man, the more we shall try by purity of life to be a living temple of the Holy Spirit, O Heart of Jesus!
The more Thy Holy Church is despised, the more we shall endeavor to be her faithful children, O Sweet Heart of Jesus!
The more Thy Vicar on earth is persecuted, the more we will honor him as the infallible head of Thy Holy Church, show our fidelity and pray for him, O kingly Heart of Jesus!
O Sacred Heart, through Thy powerful grace, may we become Thy apostles in the midst of a corrupted world, and be Thy crown in the kingdom of heaven.
Amen.
Top Act of Consecration to the Sacred Heart of Jesus

Most Sacred Heart of Jesus, I consecrate myself to Your Most Sacred Heart. Take possession of my whole being; transform me into Yourself. Make my hands Your hands, my feet Your feet, my heart Your heart. Let me see with Your eyes, listen with Your ears, speak with Your lips, love with Your heart, understand with Your mind, serve with Your will, and be dedicated with my whole being. Make me Your other self. Most Sacred Heart of Jesus, send me Your Holy Spirit to teach me to love You and to live through You, with You, in You and for You.
Come, Holy Spirit, make my body Your temple. Come, and abide with me forever. Give me the deepest love for the Sacred Heart of Jesus in order to serve Him with my whole heart, soul, mind and strength. Take possession of all my faculties of body and soul. Regulate all my passions: feelings and emotions. Take possession of my intellect, understanding and will; my memory and imagination. O Holy Spirit of Love, give me an abundance of Your efficacious graces. Give me the fullness of all the virtues; enrich my faith, strengthen my hope, increase my trust, and inflame my love. Give me the fullness of Your sevenfold gifts, fruits and beatitudes. Most Holy Trinity, make my soul Your sanctuary.
Top Litany of the Most Sacred Heart of Jesus

V/ Lord, have mercy.
R/ Lord, have mercy.

V/ Christ, have mercy.
R/ Christ, have mercy.

V/ Lord, have mercy.
R/ Lord, have mercy.

V/ Jesus, hear us.
R/ Jesus, graciously hear us.

R/ for ff: have mercy on us

God, the Father of Heaven,
God, the Son, Redeemer of the world,
God, the Holy Spirit,
Holy Trinity, One God,
Heart of Jesus, Son of the Eternal Father.
Heart of Jesus, formed by the Holy Spirit in the womb of the Virgin Mother,
Heart of Jesus, substantially united to the Word of God,
Heart of Jesus, of Infinite Majesty,
Heart of Jesus, Sacred Temple of God,
Heart of Jesus, Tabernacle of the Most High,
Heart of Jesus, House of God and Gate of Heaven,
Heart of Jesus, burning furnace of charity,
Heart of Jesus, abode of justice and love,
Heart of Jesus, full of goodness and love,
Heart of Jesus, abyss of all virtues,
Heart of Jesus, most worthy of all praise,
Heart of Jesus, king and center of all hearts,
Heart of Jesus, in whom are all the treasures of wisdom and knowledge,
Heart of Jesus, in whom the Father was well pleased,
Heart of Jesus, of whose fullness we have all received,
Heart of Jesus, desire of the everlasting hills,
Heart of Jesus, patient and most merciful,
Heart of Jesus, enriching all who invoke you,
Heart of Jesus, fountain of life and holiness,
Heart of Jesus, propitiation for our sins,
Heart of Jesus, loaded down with opprobrium,
Heart of Jesus, bruised for our offenses,
Heart of Jesus, obedient to death,
Heart of Jesus, pierced with a lance,
Heart of Jesus, source of all consolation,
Heart of Jesus, our life and resurrection,
Heart of Jesus, our peace and reconciliation,
Heart of Jesus, victim for our sins,
Heart of Jesus, salvation of those who trust in you,
Heart of Jesus, hope of those who die in you,
Heart of Jesus, delight of all the Saints,

Lamb of God, who take away the sins of the world.
R/ spare us, O Lord.

Lamb of God, who take away the sins of the world,
R/ graciously hear us, O Lord.

Lamb of God, who take away the sins of the world,
R/ have mercy on us.

Jesus, meek and humble of heart.
R/ Make our hearts like to yours.

Let us pray. Almighty and eternal God, look upon the Heart of your most beloved Son and upon the praises and satisfaction which he offers you in the name of sinners; and to those who implore your mercy, in your great goodness, grant forgiveness in the name of the same Jesus Christ, your Son, who lives and reigns with you forever and ever.

R/ Amen. Top