Q. Is there a minhag at all to mourn on the Ninth of Tamuz since that was the day when the walls of Jerusalem during the First Temple?

A. Sefer Yermeyahu (39.2, 52.6–7) states that the walls of Yerushalaim during the First Temple were breached on the 9th of Tammuz, in the year 3338 from creation (423 BCE).

The Babylonian armies of King Nebuchadnezzar breached the walls of Jerusalem on the 9th of Tammuz ; King Ziddikiahu of Yehudah was captured and taken to Babel. (Yermeyahu 39: 5). A month later, the capture of Yerushalaim was completed with the destruction of the Holy Temple and the exile of all but a small number of Jews to Babylon.

Originally, a day of fasting and mourning was observed on the 9th of Tammuz. Seventy years later, however, when the Second Temple was built, the fast was abolished and the day was turned into a holiday and a day of joy. About 500 years later when Jerusalem fell on the 17th of Tammuz — prior to the destruction of the Second Temple — the Sages decreed the 17th of Tammuz to be a fast day to commemorate both tragedies.

Tammuz 9 was observed as a fast day until the second breaching of Yerushalaim’s walls (by the Romans) on the 17th of Tammuz, 3829 (69 CE), at which time the fast was moved to that date. (Talmud, Rosh Hashanah and Tur Orach Chaim 549)
Horav Shlomo Miller Shlit’a opinion is that we don’t usually have that particular Minhag.

Rabbi A. Bartfeld as revised by Horav Yaakov Hirschman, Horav Dovid Pam, Horav Aharon Miller, Horav Chanoch Ehrentreu and Horav Kalman Ochs Shlit’a.