No. They were not already believers. There were two groups. The believers, Peter and the ones the Holy Spirit fell upon and the unbelieving Jews who were perplexed at what was happening.
They were unbelievers at the beginning of the sermon, but Peter taught four points that convinced them that they had really blundered. 1) Jesus was a man approved of God (proved by his miracles), 2) You killed him (no proof needed, they were proud of it!) 3) God raised him (proved by prophecy and witnesses), and 4) God exalted him to his own right hand (prophecy), THIS JESUS WHOM YOU CRUCIFIED.
Their reaction? v37 - when they heard this, they were pricked in their heart, and said, men and brethren, what do we do?
They were convinced, weren’t they? No longer proud of killing him, but realized they had really blundered and killed the very messiah they were watching for. They were now believers.
Thus Peter didn’t tell them to hear and believe, they already had. He told them to repent and be baptized for the remission of sins.
Nevertheless, back to my original point: your “new Christian” claimed to be such before he was baptized. Let me ask you, since you quoted Ac. 2.37-38: Do you believe this person became a Christian before he had the remission of sins? Before he was in Christ? Before he was in the body of Christ?
Thanks again for reading and considering.