‘Apostles’ in the New Testament
The term ‘priest’ described those in a special ministry but for a long time it referred to the bishop, not to presbyters. 
There were many apostles in the early days, but the Twelve had a special place in the apostolate, not so much for their missionary activity (we have no evidence that most of them left Jerusalem!) as for being the intimate companions of Jesus. Certainly in sending out the Twelve Jesus in his lifetime was signifying his intention that they participate in his present and future ministry. Luke even foreshadows a wider post-resurrection ministry with an additional sending out of the seventy (or seventy two) disciples, foreshadowing a wider group of apostles. It is precisely in the sending out of disciples (followers) that they become ‘apostles’ or ‘sent ones.’
The original apostles
Yet Luke makes it very clear that in the appointment of Matthais to succeed Judas (Acts 1:12-26) an apostle had to have ‘accompanied Jesus during all the time that the Lord Jesus went in and out among us, beginning from the baptism of John until the day he was taken up from us – one of these must become a witness with us to his resurrection.’ Quite clearly Paul misses out on the first requirement; yet Luke does not withhold the title of ‘apostle’ from Paul and Barnabas (Acts 14:4, 14). The astute reader will also note that the sermons and miracles attributed to Peter, the first among the apostles, are paralleled by Paul. For Luke, the apostleship of the Twelve is more one of par excellence than of absolute exclusivity.
Luke does not show the original apostles as missionaries and there is little evidence in Acts that the Twelve were active outside Jerusalem. It was the Hellenists and not the Twelve who spread Christianity to Samaria, for Peter and John were sent to Samaria to extend Jerusalem’s approval of a venture already well underway (Acts 8:1-15). Peter was reluctant (it took three efforts!) to accept into the Christian fold the God-fearing Gentile Cornelius (Acts 10). The great missionaries to the Gentiles were Paul and Barnabas, not the members of the Twelve as a group; they were the ones Luke shows as having the power of approval.
Who held local authority?
If the Twelve as a group were not primarily missionaries, neither were they bishops. There is no evidence in Acts that any of the Twelve presided over a local church. James, the brother of the Lord, not the apostle, was the leader of the Jerusalem church. As for Peter as bishop of Antioch, the weakness of the case was recognised when the church struck from the calendar years ago the Feb 22 feast of ‘the Chair of St Peter at Antioch’. Even Peter as the first bishop of Rome is clearly unlikely as Christianity, carried by Jewish missionaries, is known to have reached Rome in the early 40s and Peter’s arrival would not have been until the 60s. That he took over later is reading back from later church order.
The emergence of a single bishop, distinct from the college of presbyter-bishops, came relatively late in the Roman church, perhaps not until well into the second century. Leaders such as Linus, Cletus, and Clement, known to us from the early Roman church, were probably prominent presbyter-bishops but not necessarily ‘monarchical’ bishops (Ignatius’ three-tiered structure of bishop, elders and deacons). It is interesting to see the development of the tradition that made Peter the founder of the church in Rome, its bishop for one year, and in some cases 25 years, goes back to a point no earlier than the third century, and its feast to the fourth. We even have Epiphanius quoting Hegesippus that Peter and Paul were the first and joint bishops of Rome.
The probability that Peter was not the first monarchical bishop of Rome does not weaken in any way the claim that the position of primacy held by Peter has been continued in the church and is now enjoyed by the bishop of Rome. The two roles of primate and bishop of Rome, separate at first, were subsequently joined. Peter was primate among the Twelve long before he went to Rome.
Clarifying roles
The fact that the twelve apostles were not bishops or local church leaders, even coexisting in Jerusalem with James as local church leader, raises the question of what we mean by bishops being successors to these apostles. The apostolate of the Twelve and the presbyterate-episcopate (or at least local church leadership) were different roles that seemingly existed simultaneously. Clearly we have an interesting issue to explore further, one aspect of which is in the sacramental area, powers given to a select group to pass on or to the Christian community in the person of the Twelve. After all the Church recognises the right of anyone, Christian (or even non-Christian) to baptise, even though at the end of Matthew the command is addressed to the Eleven.