The words "him," "his," and "He" and refer to Paul.
# had set a day for him
"had chosen a time for him to speak to them"
# testified about the kingdom of God
Here "kingdom of God" stands for God's rule as king. AT: "told them about God's rule as king" or "told them how God would show himself as king" (See: [[rc://en/ta/man/translate/figs-metonymy]])
# from the prophets
Here "the prophets" refers to what they wrote. AT: "from what the prophets wrote" (See: [[rc://en/ta/man/translate/figs-metonymy]])
# Some were convinced about the things which were said
This can be stated in active form. AT: "Paul was able to convince some of them" (See: [[rc://en/ta/man/translate/figs-activepassive]])