ANSWER: First, let us take a look at the King James Bible verse in question in its context.
"Now about that time Herod the king stretched forth his hands to vex certain of the church. And he killed James the brother of John with the sword. And because he saw it pleased the Jews, he proceeded further to take Peter also (Then were the days of unleavened bread). And when he had apprehended him, he put him in prison, and delivered him to four quaternions of soldiers to keep him; intending after EASTER to bring him forth to the people. " (Acts 12:1 - 4, KJV)
The second part of your question needs a bit of clarification and explanation. Biblically speaking, the Passover was not a "Jewish" festival although Jews (members of the tribe of Judah or citizens of the Kingdom of Judah) certainly celebrated this Feast day. The Passover is one of God's annual Holy Days he commanded ancient Israel to keep.
The first century church did not rename the Passover to Easter. The word is a derivative of Ishtar or Astarte. It is the name of a pagan goddess of fertility celebrated during the Spring solstice.
The introduction of Sunday as a day to commemorate the death of Christ first began after Roman Emperor Hadrian crushed a Jewish rebellion (132-135 A.D.). After his victory he rebuilt the city ruins and expelled all Jews and Christians. He undertook a policy of banning the practice of any religious ceremony or event that resembled what the Jews practiced (e.g. Saturday Sabbath, etc.).
As the church historian Eusebius states, a replacement occurred of Jerusalem's Jewish-Christian leaders with Gentile church members and leaders. The new leaders changed the traditional date of observing the Christian Passover to SUNDAY. Many Gentile-Christian churches such as the one found in Rome adopted such changes. Over time, the name of the celebration changed.
The confusion concerning the word in the Bible began because scholars whose orientation was toward Catholic and Anglican church doctrines wrote the King James Version Bible. This bias caused some errors in translating the original Bible text into English. One such error was translating the Greek word pascha (Strong's Concordance #G3957), which means "Passover," as the English word Easter.
The New International Version Bible correctly translates the Greek word pascha as Passover.
"After arresting him, he put him in prison, handing him over to be guarded by four squads of four soldiers each. Herod intended to bring him out for public trial after the Passover." (Acts 12:4, NIV)
4 After the arrest, he put him in prison and assigned four squads of four soldiers each to guard him, intending to bring him out to the people after the Passover. (Acts 12:4)
The translation of Pascha into Passover, and not Easter, can be found in almost all modern reliable translations.