Cyprian’s name day is October 2, the feast of Cyprian and Justina.
Churches that follow the Old (Julian) Calendar keep the same feast on October 15 of the civil calendar. See everything commemorated on October 2.
The saint behind the name

Cyprian and Justina
October 2, Martyr, Κυπριανός και Ιουστίνη
St. Cyprian was a famed sorcerer of Antioch who was hired to bend the Christian virgin Justina to an unwanted suitor. Every art of his magic was shattered against her prayer and the sign of the Cross, and Cyprian, recognizing a power greater than all his demons, burned his books, was baptized, and in time became a bishop of the Church.
Also celebrating on October 2
These names share the feast, many of them other forms of the same name:
Kyprianos, Ioustini, Justina, Justine
What to say on a name day
The Greek wish is Χρόνια Πολλά (Chronia Polla), “many years.” A call or a short message on the day is what people remember. See name day wishes you can send and what a name day is.