Why didn't the Apostle Paul mention Mary and her feast and holy days?
The writings of Paul that were preserved to be included in the New Testament were inspired by the Holy Spirit and presumably, the Holy Spirit did not think any mention of Mary was needed in Paul's writings.
To the best of our knowledge there were no saint feast days during the lifetime of Saint Paul. Although I wonder if Paul remembered the martyrdom of Saint Stephen (which he whitnessed and possibly participated in) every year.
And, just like the doctrine of the Holy Trinity, the significance of the Blessed Virgin Mary was not understood until the Holy Spirit guided the Church toward it a few generations later.
That said, the Blessed Virgin Mary is important to Catholics primarily because the Bible says she is important to God.
• Mary is the handmaid of the Lord (Luke 1:38), a servant of God just like us.
• Mary is the first Christian. All other Christians follow her on the road of faith. She is the model for all of us because she actively cooperated with the Will of God by saying "Yes" to God in the plan of salvation.
• Mary is the model of faith. Mary's faith never wavered throughout Jesus' life, even while watching him die on the cross when most of the Apostles ran away.
• Mary is full of grace. The angel Gabriel called her, "Full of grace" and said, "The Lord is with you." She is full of grace at that moment; the Lord is with her at that moment, before she says "yes" to the angel's question. Catholics believe the state of grace was with her since her Immaculate Conception. God prepared her for her later role as the mother of Jesus.
• Mary is the mother of Jesus Christ, of God the Son.
• Mary is a dwelling place of the Holy Spirit, a tabernacle. The angel Gabriel says to her, "The Holy Spirit will come upon you and the power of the Most High will cover you with its shadow." (Luke 1:35)
• God praised Mary through His messenger, the angel Gabriel, "Rejoice, Mary, full of grace, the Lord is with you." If God and his angels praise Mary, shouldn't we?
• Jesus honored Mary. Jesus kept the Commandments and did not sin. Jesus fulfilled the Commandment to honor both his father and his mother. WWJD? What would Jesus do? Jesus would and did honor Mary, his mother.
• The Holy Spirit inspired praises of Mary. When Elizabeth heard Mary's greeting, the infant leaped in her womb, and Elizabeth, filled with the Holy Spirit,
cried out in a loud voice and said, "Most blessed are you among women," and "Blessed are you who believed that what was spoken to you by the Lord would be fulfilled." (Luke 1:41-45)
• The Scriptures tell of Mary's place with Jesus. Mary gives birth to Jesus, presents him to the shepherds, to the wise men, to God in the Temple. She lived with him for many years in Nazareth, intercedes with Jesus in Cana, suffers with him at the foot of the Cross, and prays to him with the apostles in the Upper Room.
• Mary was a prophet. "Behold, from now on will all ages call me blessed. The Mighty One has done great things for me." (Luke 1:48-49) As Christians who believe the Bible, we must call Mary "blessed."
• Mary is our Mother too. While he was on the Cross, Jesus said to his beloved disciple (who represents all Christians), "this is your mother." (John 19:27)
For more information, see the Catechism of the Catholic Church, section 484 and following.
With love in Christ.