“On Rosh HaShana, which is coming up, we don’t cry … sweet new year.”
The Gr”a says that it’s technically prohibited to cry publicly on R.H. because it is Yom Tov (& Shabbos when applicable).
But it should be known that the Arizal says that if a person is not moved emotionally to tears on R.H., either from the Tefillos (e.g., Avinu Malkenu, Un’tana Tokef), or from the Tekios, etc…, then that person’s neshama is “flawed” and needs real work/repair.