This passage is not about Paul appointing bishops, per se, but that they were there in the early Church.
Matthew 28:18-20
"And Jesus came and spake unto them, saying, All power is given unto me in heaven and in earth. Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost: Teaching them to observe all things whatsoever I have commanded you: and, lo, I am with you always, even unto the end of the world. Amen."
The Apostles went to all nations: James the Greater to Spain, Thomas to India, etc. They appointed bishops to succeed them, especially since they (the Apostles) were all martyred within a relatively short time. (There are articles on-line which explain which Apostles went where and how they were martyred.) The only one who was not was St. John, who had been exiled to the island of Patmos and lived to be quite elderly.
The bottom line is that there are historical documents out there which describe the Early Church in detail. You just have to be willing to research.
St. John is actually a martyr. He was boiled in oil, but was left totally unharmed. Because he was willing to die for the Church and endured something that should have killed him, the Church has always considered him a martyr.