E. G. Richards says in his book, Mapping Time:The Calendar and its History that the Jews adopted the Babylonian calendar during the Captivity. This was lunisolar: each year had 12 lunar months and a 13th was intercalated as necessary. The year started in the autumn and the day at sunset and the months were deemed to begin when the new moon was first observed at Jerusalem. He says the Jews used a seven-day week of great antiquity possibly of Babylonian origin, but the days were not named, just numbered – except for the seventh day, the Sabbath. The Roman calendar was originally lunar; kalends represented the first day of a month and originally marked the appearance of the new moon. By the first century, Julius Caesar had reformed the year by creating the Julian calendar of 365 days and a leap day every fourth year, effectively making this a solar calendar.
I assume your reference to "the author of John's Gospel" is an acknowledgement that the author was not really the apostle John. This is the consensus of New Testament scholars, as noted for example in The Birth of Christianity, pp 20-21, by John Dominic Crossan. John's Gospel was written in Greek and almost certainly not in Palestine, but the author seems more aware of Jewish customs than were the authors of the synoptic gospels. When he altered the time of the crucifixion from the third hour on the day following the seder feast, to the sixth hour on the day before the feast (John 19:14), he was aware that this was when the lamb was sacrificed according to second-temple Jewish ritual.