When is the first documented case of a Christian making a confession of his sin to the priest/bishop/elder with a purpose of having it remitted?
The earliest confessions seem to have taken place in a communal setting, before both whoever was in authority and the wider group simultaneously. The first documented examples are probably the people who came to John the Baptist (first half of first century CE), confessing their sins publicly and being baptised for the purpose of "repentence", which implies remission of sins was the goal (Matthew 3:6, 11; Mark 1:5; cf. James 5:16). But you may not consider them "Christian" - if not, then I'd point to the new Christians of Ephesus. As a result of Paul's preaching, they gathered together publicly, confessed their sins and burned their books of sorcery in a demonstration of their repentance (Acts 19:18-19).
The earliest post-NT sources that discuss confession of sins to a human (as opposed to God) are also vague; two of the earliest appear to be by the Didache and Clement of Rome (written between the mid-1st century and mid-2nd century CE).
The Didache (14) instructs its readers to confess their sins in the congregation (ecclesia) when they meet on "the Lord's Day". It doesn't explicity mention confession for remission of sins, but the purpose of confession is so that the confessor may be "pure" when breaking bread (called the "sacrifice", i.e. the eucharist), so forgiveness of sins as a result of confessing to the congregation is implied.
Clement of Rome (Corinthians 57) instructs the wicked to repent and submit to "the presbyters" and receive "correction." Again, confession is not explictly mentioned as a part of this process, but is implied.
More detail on other patristic sources on the issue, see the Catholic Encyclopedia article on the sacrament of penance.
If you're asking for a specific individual, that's a bit more difficult, since examples of individuals are rarely given (unless they are members of a heretical movement). The earliest is probably the nameless "young man" cited by Clement of Alexandria in "Who is the Rich Man That Shall Be Saved?" (written early third century CE). In his tale, which may be purely legendary, the apostle John, after the Emperor Domitian's death (96 CE), left his exile on the island of Patmos and returned to Ephesus. A young man who had been baptised fell in with a bad crowd and became a highway robber, and tried to rob John before he recognised the apostle and fled. John pursued and cried out,
“Why, my son, do you flee from me, your father, unarmed, old? Son, pity me. Fear not; you have still hope of life. I will give account to Christ for you. If need be, I will willingly endure your death, as the Lord did death for us. For you I will surrender my life. Stand, believe; Christ has sent me.”
And he, when he heard, first stood, looking down; then threw down his arms, then trembled and wept bitterly. And on the old man approaching, he embraced him, speaking for himself with lamentations as he could, and baptized a second time with tears, concealing only his right hand. The other pledging, and assuring him on oath that he would find forgiveness for himself from the Saviour, beseeching and falling on his knees, and kissing his right hand itself, as now purified by repentance, led him back to the church. Then by supplicating with copious prayers, and striving along with him in continual fastings, and subduing his mind by various utterances of words, did not depart, as they say, till he restored him to the Church, presenting in him a great example of true repentance and a great token of regeneration, a trophy of the resurrection for which we hope... For he who in this world welcomes the angel of penitence will not repent at the time that he leaves the body, nor be ashamed when he sees the Saviour approaching in His glory and with His army.
The event involves weeping, declamations, lamentations, prayer, fasting, repentance and forgiveness, between the penitent and the apostle. This is of course a far cry from the medieval or modern practices of confession, repentance and penance under the auspices of church authorities, but those practices evolved over centuries to the familiar model we see today.