Acts 12 does not say or imply that Peter was imprisoned in Jerusalem in 44 AD.
-A8
As far as the year 44 AD, it must be interpolated from Scripture.
Here is what the scripture does say: [Acts 11:28] 28 One of them, named Agabus, stood up and through the Spirit predicted that a severe famine would spread over the entire Roman world. (This happened during the reign of Claudius.
Continuing in Chapter 12: [Acts 12:1-5] 1 It was about this time that King Herod arrested some who belonged to the church , intending to persecute them. 2 He had James, the brother of John, put to death with the sword. 3 When he saw that this pleased the Jews, he proceeded to seize Peter also. This happened during the Feast of Unleavened Bread. 4 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. 5 So Peter was kept in prison, but the Church was earnestly praying to God for him.
Claudius...."Tiberias Claudius" reigned from 41 A.D. to 54 A.D. so we now know that Peter was imprisoned sometime after 41 A.D. and escaped [Acts 12:8] with divine intervention shortly thereafter. (verse 11)
King Herod died in 44 A.D. so Peter had been in the local pokey sometime between 41 to 44 A.D.
When do you think he was there?