In a place where there are not recognized tzaddikim per se, is there still an inyan to go and daven at the gravesite of family, friends, or communal leaders, etc?
Regarding this specific minhag, is there still inyan to go earlier in the month?
The main minhag is to go to kevarim of tzadikim and there are poskim who say that it is only to the kever of tzaddikim. On the other hand, it is hard for us to say who is considered a tzaddik. There are many people from the previous generation that had tremendous nisyonos, and passed them, they may not be recognized, famous, tzaddikim, but they are tzaddikim nonetheless. Just as an example, the Satmar Rebbe zt”l used to say to people who came to him for a bracha, ” Why are you coming to me, just find a yid with a number tattooed to his forearm (a symbol that he went thru the concentration camps) and ask him for a bracha”. Additionally, there were many people who were moser nefesh in the previous generations just to keep Shabbos, or even just to stay frum, and bring up a frum, torah family. They look like regular “simple” people, but they are tzaddikim nonetheless. Additionally, local Rabbonim, etc would also be considered tzaddikim.
The idea of going to the kevarim is to daven because tefillos are accepted better over there, since it is a holy place. The main idea is to daven that H-shem should give us a good year- that can also be done earlier if it can’t be done on Erev Rosh Hashana.
O:CH 581-4, M:B 581-27, Kovetz Halachos ( Yomim Noraim) 4-7 ftnt. 12