I would only add a slight correction - Jesus did not mean the Old Testament as we know it. We do not know which Scriptures He meant, specifically. Or which textual versions of which Scriptures.
That’s not really accurate - Christ specifically refers to “all the books of the Law and the Prophets”, which would mean in the context of Judaism the five books of the Pentateuch and various other books in the Old Testament.*
Additionally, there are clear prophecies of our Lord in the Old Testament, a great many of which were explicitly cited by St. Matthew in his Gospel, and others of which were called to mind in the Gospel of St. John, for example, its recapitulation of John 1:1. And there are numerous direct quotations of the Old Testament throughout the New.
Finally, what I have said regarding this, that the Old Testament is Christological prophecy, is the Patristic view, the view of traditional liturgical churches such as the Orthodox, Catholics, Protestants and Assyrians (it is for this reason the ancient lectionaries such as the East Syriac, Mozarabic, Gallican and Ambrosian lectionaries, the Roman Rite lectionary and its Protestant derivatives, and the Byzantine, Coptic, Syriac, Ethiopian and Armenian lectionaries, as well as the Revised Common Lectionary and the other recent three year lectionaries such as that of the Novus Ordo Missae will preface, in one way or another, New Testament lessons with Old Testament lessons, either in the Eucharistic liturgy itself or in the Divine Office (for instance, Eastern Orthodox and Oriental Orthodox churches tend to reserve the Divine Liturgy for the reading of the New Testament, and read the Old Testament prophecies that pertain to the liturgical occasion the night before at Vespers).
Many of the most obvious Christological prophecies pertaining specifically to the Passion and Resurrection of Christ are read on Holy Saturday morning at the Vesperal Divine Liturgy in the Eastern Orthodox church, where 14 lessons are read, or the Paschal Vigil Mass of the Roman Rite in its pre-1954 form** where 12 lessons are read; these exist to be read during the baptism of catechumens, and historically additional prophecies would be read as needed until the baptisms were complete, thus preparing the baptized neophytes for hearing the Gospel and partaking of the Eucharist at the Paschal liturgy.
* It is true, however, that the early Church never agreed exactly on the contents of the canon of Old Testament scripture, with variations existing between the various ancient liturgical rites and even slight variations within the Byzantine Rite as practiced by Eastern Orthodox of the Mediterranean and the Byzantine Rite as practiced by Eastern Orthodox speaking Church Slavonic, such as the Serbians, Russians, Bulgarians, Ukrainians, etc. And additionally there is the issue of the divergent Old Testament used by the Ethiopians and Eritreans before and after their conversion, which has never been enough of an issue for the Coptic Orthodox Church to try to act to suppress it, but which does include books such as 1 Enoch and Jubilees not found elsewhere, although St. Jude in his epistle appears to quote 1 Enoch. In the Protestant reformation, a number of churches adopted the 66 book Masoretic canon, but the Anglicans did not, nor did John Calvin, who apparently regarded Baruch as canonical, nor really Martin Luther, who would have preferred Esther be removed from the canon (it is my understanding that Lutherans have an open canon with regards to the Old Testament). However, it must be stressed that these differences are
minor, for the most important prophetic texts exist across
all canons of Old Testament Scripture.
** Before the 1954 reforms of Pope Pius XII, Paschal Vigils was celebrated in the morning, and some traditional Latin mass groups within the Roman Catholic Church, but not the SSPX, follow this practice; additionally, ,the reforms of Pius XII also changed the Mass of the Presanctified so that it no longer resembled the Presanctified Liturgy used by the Eastern Orthodox and Byzantine Rite Catholics, for both liturgies were based on a revised presanctified liturgy composed by Pope St. Gregory I Diologos, also known as St. Gregory the Great, among his numerous other liturgical contributions, such as Gregorian chant.