Is there any information about the calendar system used in Carthage ? Was it solar or lunar ? What date did Carthaginians took for a reference day (1st day of the 1st year) ?
The Carthaginians were culturally Phoenician, and most evidence I've come across points to the calendar being lunisolar. There are some pretty strong indications that it would have been similar to (or evolved into) the Hebrew calendar, and there are several month names that are shared with other cultures in the region. Phoenician feasts and rituals revolve around the harvest, which would likely point toward marking the new year on the spring equinox. See Goldfarb and Markoe for more information.
The lunisolar calendar began in the first iteration of the city-state Babylon, when the sun and moon were still synchronized to 30 days. It has, obviously, de-synchronized since. The queen of Babylon used the moon-based calendar to given names to the months so that her family would be each identified with a heavenly body, with herself as the moon. The calendar that she replaced was the one that was used in Byblos, a city that was already old when Babylon was built. It is strictly sun-based, and was defined in the writing called Book of Enoch that has subsequently been edited as earth changes dictated. The last and most profound edit caused the book to be re-titled The Book of Jubilees, because it intercalated two weeks, one in the spring and one in the autumn, along with other adjustments to be aligned with the actual path of the sun. There were 14 copies or partial copies of this calendar found in the caves in Qumran, all written 200 years before the Romans destroyed Jerusalem. It was still used to set the temple feasts by the priests when the temple was destroyed by the Romans. The moon-based calendar was forced on the nations that were captured first by Babylon, then by the nations that rose from the Babylonish after-effects of their captivity, including Egypt. Ethiopia still has a sun-based calendar similar to the adjusted one, but it is only used for calculating their Christian church days of festivals. The Jubilee, or Qumran calendar, as it is variously called, is not in use by any known political unit today, due to the Roman political system being adopted world-wide. The Hebrew prophets said that it would be re-established when a king is anointed over the regathered 12 tribes in the end times.
The Carthaginians used the African calendar, also known as the "Coptic Calendar", which was based on the era of Nabonassar, which is the Babylonian calendar as adopted from the Chaldeans. The months of the ancient African calendar are as follows:
Each month had 30 days plus there were 5 or 6 epagomenal days depending on the year which were placed at the end of the year.
The Punic Thoth 1 was on the same day as Thoth 3 in the era of Nabonassar, thus was 2 days in front of the regular Babylonian calendar. The year 546 in the era of Nabonassar and the day Punic Thoth 1 corresponds to our day October 15, 203 B.C.
The African calendar was a solar calendar and a variant of it is still in use by the Coptic church. In ancient times, this same calendar was used throughout north Africa, including Libya, Egypt and Ethiopia.