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 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.