Amvrosios’s name day is December 7, the feast of Ambrose of Milan.
Churches that follow the Old (Julian) Calendar keep the same feast on December 20 of the civil calendar. See everything commemorated on December 7.
The saint behind the name

Ambrose of Milan
December 7, Hierarch, Αμβρόσιος
St. Ambrose was Archbishop of Milan in the fourth century and one of the great Fathers of the Western Church. Governor of the region when a disputed episcopal election threatened the city with strife, he was acclaimed bishop by the people while still a catechumen; he gave his wealth to the poor, defended the Nicene faith against Arianism, and did not fear to call even emperors to repentance. His preaching helped bring Blessed Augustine to conversion.
Also celebrating on December 7
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.