It is not altogether clear that the OP quoting Cyprian actually supports his question. After all, Cyprian was writing against Novatian, who was a pope (anti-pope by others). So, when Cyprian says he was arguing against priests, it was a call to unity in the Church, rather than a clear assertion of sacerdotalism. But let us proceed with answering the question.
Sacerdotalism means these things, but is primarily concerned with a priest offering a sacrifice and a sacrifice requiring a priest.
Sacerdotalism is the belief that priests are essential mediators between God and man. The word sacerdotal comes from the Latin word for “priest” and literally means “to make sacred.” Catholic, Orthodox, and Anglican forms of worship are sacerdotal.
Sacerdotalism teaches that the priesthood is a special class of churchman and is a necessary part of worship. People cannot approach God on their own, but must come through a priest, whether it is for confessing sin, taking communion, or receiving grace. Any divine blessing conferred upon an individual comes through the Church; that is, through the Church’s ordained priesthood. GotQuestions
Sacerdotalism (from Latin sacerdos, priest, literally one who presents sacred offerings, sacer, sacred, and dare, to give)1 is the belief in some Christian churches that priests are meant to be mediators between God and humankind. Wiki
As regards the person offering, it is not permitted that any and every individual should offer sacrifice on his own account. In the revealed religion, as in nearly all heathen religions, only a qualified person (usually called priest, sacerdos, lereus), who has been given the power by commission or vocation, may offer up sacrifice in the name of the community. NewAdvent
OP: How did the early church understand the priesthood?
As mentioned also, the early church understood the priesthood not as an assigned office, but as an inherited position. Inherited in the sense of being automatically adopted into by virtue of our being born-again into Christ.
Peter, ironically enough, calls believers priests.
Ye [the redemeed] also, as lively stones, are built up a spiritual house, an holy priesthood, to offer up spiritual sacrifices, acceptable to God by Jesus Christ. 1 Peter 2:5
John too calls us priests.
And hath made us kings and priests unto God and his Father; to him be glory and dominion for ever and ever. Amen. Rev 1:6
Paul in detailing the Passover instructs believers. There is no sense of a separate class of priests.
For I have received of the Lord that which also I delivered unto you [brethren, verse 2], That the Lord Jesus the same night in which he was betrayed took bread: 1 Cor 11:23
OP: Did the apostles see themselves as priests?
In short, no, they did not see themselves as priests, elders yes, but not priests like the OT Levitical priest. As New Advent clears up, the word presbyter came to mean priest. In other words, it was not how it was first defined.
This word (etymologically "elder", from presbyteros, presbyter) has taken the meaning of "sacerdos", from which no substantive has been formed in various modern languages (English, French, German). The priest is the minister of Divine worship, and especially of the highest act of worship, sacrifice. Priest
Lastly, the redemptive sacrifice of Christ was offered once for He then sat down after offering His own blood and this Priest never offering the blood of others.
But this man [Christ], after he had offered one sacrifice for sins for ever, sat down on the right hand of God; Heb 10:12
Nor yet that he [Christ] should offer himself often, as the high priest entereth into the holy place every year with blood of others; Heb 9:25
So, in conclusion, the early church saw every believer as a king/priest offering bread/wine and sacrifice of praise modeled after Christ Jesus with church elders officiating.