Anyone can baptise, provided they use the right formula of words, have the intention to do as the church requires, and performs the proper actions.
Paul’s main mission was to preach the gospel, this is correct, Jesus sent him for the proclamation of the Faith.
Remember when John the Baptist came along and said this,
Matt. 3:11
"I indeed baptize you with water unto repentance. but he that cometh after me is mightier than I, whose shoes I am not worthy to bear: he shall baptize you with the Holy Ghost, and with fire:"
This is the spiritual baptism of Colossians 2 and Romans 6. It's the circumcision of Christ, the circumcision made without hands.
It's what we call the born-again experience.
You've probably heard the old preachers especially in the 1800's preach on it. This is before the Pentecostal Movement came around 1900 and got it all mixed up with their baptism of the Holy Spirit, which is entirely a different baptism.