"Dear Prof. Foster,
Which are the first episodes of prayers addressed to Christ?"
In answering your question, I will assume that prayer to Christ was not a first century practice. Even certain systematic theologians like Owen Thomas have pointed out that traditional Christian prayer follows the schema of "to the Father, through the Son and in [i.e. through] the holy spirit." See Thomas' _Theological Questions: Analysis and Argument_ (Wilton, Connecticut: Morehouse-Barlow, 1983. P. 71).
Ignatius of Antioch, who was martyred between the years 98-117 CE, may possibly indicate that Christians invoked Jesus Christ in his time (To the Ephesians 20). However, one could no doubt understand his words as a reference to believers praying to the Father through the Son qua High Priest and Intercessor. While the middle recension of the Ignatian letters evidently does not contain the reading that explicitly speaks of praying to God through Christ, another recension does speak of the Son interceding for worshipers of God in his capacity as High Priest. At any rate, Ignatius of Antioch may well have advocated or espoused prayer to Christ.
Some slightly ambiguous evidence is also available for us in the Epistles of Pliny the Younger, who is known for writing to the Roman Emperor Trajan in 112/113 CE regarding Christians in Asia Minor, who would sing psalms to the Lord Jesus Christ quasi Deo. Did such hymns also include prayers to the Son of God? It is possible that this may have been the case, though I don't think one can say for certain, based on Pliny's letters alone (see the letters of Pliny X.96.7).
Origen too addresses the subject of prayer in De Oratione 15.1. There, he insists that, properly speaking, prayer should be addressed to "God the Father alone," although he also invokes Christ, "the very Logos himself." Nevertheless, in Contra Celsum 5.4, Origen demonstrates that he does not believe prayer in the absolute sense should be directed toward God's Son. See Jaroslav Pelikan's _The Christian Tradition_ 1:198-199.
You might also want to reference Arnobius of Sicca's Adversus nationes 1.36.
I hope these thoughts are helpful.