Q. Is one supposed to be happy on Rosh Hashana or worried because it is the Yom Hadin? Does one greet his friend with Gut Yom Tov on Rosh Hashana?

A. See above question 1437. In principle a person should worry and fear the Yom Haddin, since it is the judgment day when everyone’s fate and destiny is being decided. Yet, at the same time, he should place his trust in Hashem’s compassion and help to those who decide to better their ways. Talmud (Taanis 26b, 30b) mentions that the greatest Yom Tov ever was Yom Kippur itself, since it was a day of forgiveness and absolution.
Horav Shlomo Miller’s Shlit’a opinion is that it is proper to greet people with a gut Yom Tov on Rosh Hashana. And that seems to be the tradition of many but not all.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit’a