Makary’s name day is January 19, the feast of Makarios of Egypt.
Churches that follow the Old (Julian) Calendar keep the same feast on February 1 of the civil calendar. See everything commemorated on January 19.
The saint behind the name

Makarios of Egypt
January 19, Venerable, Μακάριος
St. Makarios of Egypt, called the Great, was one of the foremost of the desert fathers of the fourth century. Retiring to the desert of Scetis, he became the father of the monks who gathered there, renowned for his humility, his gentleness toward sinners, and his deep experience of prayer.
Also celebrating on January 19
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.