The most commonly celebrated Name Day for Greeks with the name Anastasios and Anastasia is Greek Easter, as the name means "resurrection." This is a moveable feast day, falling on a different Sunday each year.
I studied Koine Greek in college, and did the etymology of my first name. Ana is a preposition which means 'back up' or 'up again', and stasis is the future form of the verb histemi, which means 'to stand' or 'to rise'. So the literal meaning of the name is 's/he who shall rise up again.' It was apparently given to children born around Easter during the early years of Christianity. I pronounce it ah-nah-stah-SI-Ah, which is the pronunciation used in Holland and Russia.
Yes, this name is also pure Russian. It is spelt Anastasiya in Russia, but this name is Russian, no matter how it is spelt. Also, a common pet name for Anastasia is Anya.