What did early Christians say about apostolic succession? Did early Christian writers teach apostolic succession or reject it? How do their teachings on this topic compare with contemporary and significant historical understandings of apostolic succession?
To answer the question, no the early church did not reject "apostolic succession".  But to understand the answer, it is necessary to define exactly what the question is; what precisely is "apostolic succession"?  For the early church, it meant "teach the same as apostles".  It did not mean, what it came to be later redefined as, some sort of sacerdotal succession.

CLASSIC APOSTOLIC SUCCESSION

The classic (earliest) definition of apostolic succession was to "teach the same" as apostles.  Thus you might find apostolic succession in a church from Judea to Great Britain to India; all teaching the same as the apostles.  It was the binding to "the faith once delivered".


  
  It is within the power of all, therefore, in every Church, who may wish to see the truth, to contemplate clearly the tradition of the apostles manifested throughout the whole world; and we are in a position to reckon up those who were by the apostles instituted bishops in the Churches, and [to demonstrate] the succession of these men to our own times; those who neither taught nor knew of anything like what these [heretics] rave about
  Irenaeus AH Book III, III, 1 (bold mine)
  


In other words, Irenaeus is contrasting the false teachers who taught things unknown with and apart from what apostolic churches were teaching, which was handed down in succession.  There was no room for innovation or tradition.


  [Polycarp] departed this life, having always taught the things which he had learned from the apostles, and which the Church has handed down, and which alone are true.
  -ibid, 4


Again, the definition of "apostolic succession" is detailed as handing down that truth which was learned from apostles.  Those teachings alone are true.

REDEFINED APOSTOLIC SUCCESSION

The Catholic Church (CC) now defines apostolic succession in this way.


  1399 The Eastern churches that are not in full communion with the Catholic Church celebrate the Eucharist with great love. "These Churches, although separated from us, yet possess true sacraments, above all - by apostolic succession - the priesthood and the Eucharist, whereby they are still joined to us in closest intimacy." A certain communion in sacris, and so in the Eucharist, "given suitable circumstances and the approval of Church authority, is not merely possible but is encouraged.
  -Catechism CC-


In other words, even though Catholics and Orthodox differ on major theological doctrines, CC may still define itself and Orthodox as having "apostolic succession" by virtue of a claim to a type of sacerdotal priesthood.

There's no need to delve into which denomination accurately teaches what the apostles taught because the point is made; that is, "apostolic succession" has been redefined from "teach the same as apostles" to having some type of "physical continuity as priests".

SUMMARY


  
  Wherefore it is incumbent to obey the presbyters who are in the Church,—those who, as I have shown, possess the succession from the apostles; those who, together with the succession of the episcopate, have received the certain gift of truth, according to the good pleasure of the Father. But [it is also incumbent] to hold in suspicion others who depart from the primitive succession, and assemble themselves together in any place whatsoever, [looking upon them] either as heretics of perverse minds, or as schismatics puffed up and self-pleasing, or again as hypocrites, acting thus for the sake of lucre and vainglory. For all these have fallen from the truth. 
  Irenaeus Book IV, XXVI, 2
  


Somewhere along the historic line, the shift was made from adhere to those who teach the same apostolic truth to obey your priest and bishop who can offer a continuous sacrifice for you.

To have "apostolic succession" as it was defined and practiced in the earliest Christian years, one simply must "teach the same" as apostles.