If you assume Acts is history, then you'd date the First Epistle to the Corinthians to 53 - 57 AD, however, as Dennis R MacDonald has pointed out, the Acts of the Apostles actually seems to imitate the writings of Homer. Trying to find out about the historical Paul by reading Acts is the same as trying to uncover the historical Achilles by reading the Iliad.
While there is some historical truth to the Iliad (there was a city called Troy which was destroyed in a war, some of the characters were actual historical figures) that doesn't mean the Iliad is history. Likewise, while Acts contains some historical truth (many places and people mentioned did indeed exist), that doesn't mean Acts is history. Acts is a historical romance novel and we must treat it that way.
So if we abandon the chronology of Acts, when was 1 Corinthians written? The earliest papyrus of 1 Corinthians dates to about 200 AD and we know it was included in the Apostolikon of Marcion (c. 130 AD). We have no evidence of its existence before this. Turning to the text itself, 1 Corinthians mainly deals with second century gnostic ideas, especially those of the Cerinthians.
In fact, it's more likely Corinthians was written with the Cerinthians in mind than the inhabitants of Corinth. There is no archeological or historical evidence of Jews (let alone Christians) living in the Roman city of Corinth until late in the first century. It's highly unlikely for there to have been a Christian church in Corinth for Paul to have written to during his lifetime.
1 Corinthians makes a lot more sense as a second century response to the teaching of the Cerinthians, a gnostic group that practiced baptism for the dead (according to Epiphanius of Salamis, Panarion 28.7). Cerinthians kept the Jewish law including the practice of circumcision and distinguishing between clean and unclean food. They believed celibacy was preferable to marriage. They denied that Jesus rose from the dead, and they had no qualms about eating meat that had been used to sacrifice to idols. 1 Corinthians addresses baptism for the dead (1 Corinthians 15:29), whether Jesus was resurrected (1 Corinthians 15:12), circumcision (1 Corinthians 7:19), celibacy (1 Corinthians 7) and eating sacrificial meat (1 Corinthians 8, 10:18-33).
In The Pre-Nicene New Testament, Robert M. Price supposes that the title of the epistle Corinthians was meant to be a pun on Cerinthians. Whoever compiled it couldn't just title it Cerinthians since it was supposedly written before Cerinthus was born, but second century readers would know to which group it was really directed. Just like the Book of Mormon which was supposedly written in ancient times, but addresses nineteenth century issues, 1 Corinthians is a wholly second century document masquerading as a letter from the first century.