Exposition and Adoration is held every First Friday of the month from after 12:10 mass until Benediction which preceeds 5:10 Latin Mass.

Catechism of the Catholic Church, Art. 1380 

It is highly fitting that Christ should have wanted to remain present to his Church in this unique way. Since Christ was about to take his departure from his own in his visible form, he wanted to give us his sacramental presence; since he was about to offer himself on the cross to save us, he wanted us to have the memorial of the love with which he loved us "to the end," even to the giving of his life. In his Eucharistic presence he remains mysteriously in our midst as the one who loved us and gave himself up for us, and he remains under signs that express and communicate this love:

The Church and the world have a great need for Eucharistic worship. Jesus awaits us in this sacrament of love. Let us not refuse the time to go to meet him in adoration, in contemplation full of faith, and open to making amends for the serious offenses and crimes of the world. Let our adoration never cease.

Every First Friday of the month, Cathedral offers a Latin Novus Ordo Mass at 5:10 pm after Adoration and Benediction.  The Novus Ordo Mass is the same as the Ordinary form of the Mass, just that parts are spoken and sung in the Latin language.  Booklets are provided with everything needed to particpate.  The Second Vatican Council encouraged the use of Latin in certain parts of the Mass, although exposure to Latin is not common today. 

All the faithful should be able to sing or say in Latin the parts of the Mass which concern them (Sacrosanctum Concilium, Art. 54) 

This Mass is currently not the Extraordinary form or the Tridintine Mass.

http://www.ewtn.com/library/LITURGY/VENDLATN.TXT