Polykarpos’s name day is February 23, the feast of Polykarpos.
Churches that follow the Old (Julian) Calendar keep the same feast on March 8 (a day earlier in leap years) of the civil calendar. See everything commemorated on February 23.
The saint behind the name

Polykarpos
February 23, Hierarch, Πολύκαρπος
St. Polycarp was Bishop of Smyrna and a disciple of St. John the Theologian, a living link between the apostles and the generations that followed. He shepherded his church for many decades, and in extreme old age was arrested during a persecution in the second century. Urged to deny Christ, he answered that in eighty-six years the Lord had done him no wrong, and he could not blaspheme his King.
Name days: Polykarpos, Polycarp
Also celebrating on February 23
These names share the feast, many of them other forms of the same name:
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.