Q: Are there any mitzvot that a Noahide should perform on the Ninth of Av? Should he read the Book of Lamentations, fast, or mourn? If so, what are the specific instructions?
A: There are no commandments binding on a Noahide on the Ninth of Av. Any actions that he or she takes are completely voluntary. Anybody who decides to fast, or to read the Book of Lamentations or the Wars of the Jews and the Romans by Josephus, will be blessed for compassion.
The same is true of other restrictions, such as not listening to music, not greeting other people, and not using makeup. All such practices are copied from the obligations of Israel and are voluntary for Noahides. Children should not be told to fast.