Well, if you read the Bible again, this time more carefully, you will notice that Peter declares "let his bishopric another take" to refer to Judas and the need for someone to succeed him in his episcopate since he died.
Clement (mentioned by Paul in his letter to the Philippians), who was one of the earliest bishops in Rome, wrote a letter in the first century which explains that the Apostles ordained certain men to succeed them in their episcopate once they had died.
Bishopric is a weak translation in Acts 1:20. In a few older English translations the Greek word, episkope is transliterated as bishopric. Most translate it rather than transliterate it and render it as in the following:
20 For it is written in the book of Psalms,
LET HIS HOMESTEAD BE MADE DESOLATE,
AND LET NO ONE DWELL IN IT;
and, LET ANOTHER MAN TAKE HIS OFFICE.
20For it is written in [the] book of Psalms, Let his homestead become desolate, and let there be no dweller in it; and, Let another take his overseership.
"For it is written in the Book of Psalms,
"'May his camp become desolate,
and let there be no one to dwell in it';
and
"'Let another take his office.'
20for it hath been written in the book of Psalms: Let his lodging-place become desolate, and let no one be dwelling in it, and his oversight let another take.
20for it hath been written in the book of Psalms: Let his lodging-place become desolate, and let no one be dwelling in it, and his oversight let another take.
In this verse Luke quotes Psalm 109:8 (or 108:8 in Catholic Bibles). It is variously translated as follows:
8Let his days be few; and let another take his office.
8His days are few, his oversight another taketh,
8Let his days be few, let another take his office;
8May his days be few;
may another take his office!
The point is that Matthias was chosen to replace Judas. The office, or oversight, that Judas held was that of an Apostle. In no other place in scripture are Apostles equated with overseers (elders, bishops, persbyters).