1) It is customary [for men] to immerse in a kosher Mikvah on Erev Rosh Hashana, regardless if they are impure or not, in order to spiritually purify themselves in preparation for the holy day. The accepted custom is to immerse in the Mikvah three times to signify the 3 times it says the word "Tahara- purity" in the Pasuk "V’Zarakti Aleichem Mayim Tehorim. U’Tehartem MiKol Tumoseichem U’Mikol Giluleichem Ataher Eschem- [Hashem says] I will sprinkle purified water upon you, and purify you from all your impurities; and from all your filthiness I will purify you" (Yechezkael 36:25)(Sefer Chasidim, Mateh Ephraim and others). There are other Minhagim regarding how many times to immerse. Some do it once, some do it 7 times, others do it 12 times etc. If you don’t have an established personal custom regarding this, rely on the above and do it three times.
The earliest time for going to the Mikvah is an hour before Chatzos (which is around 1:00pm See http://www.myzmanim.com/ for exact Halachic times in your specific location). 2) One should constantly have "Teshuva" on his/her mind throughout the day on Erev Rosh Hashana (Chayei Adam)
Many have the Minhag to go to a grave of a Tzaddik and daven there that Hashem should have mercy on us and forgive our sins and grant us a good year in the merit of the Tzaddik. If a no Tzadik’s grave is accesible, the Poskim recommend going to any Jewish cemetery to daven. (Some even say that if no Jewish cemetery is available, one should go to the outside of a non Jewish cemetery, as just going to any cemetery reminds us that we will not live forever, and that we better start doing Teshuva before it’s too late)
One who is impure (due to nocturnal emissions) should preferably not visit a cemetery, especially on Erev Rosh Hashana, as it can be harmful to him. (The Gaon of Vilna and others were extremely stringent about this). It is also better to visit the cemetery before eating, thus many have the Minhag to go immediately after Shacharis.
It is a good custom to give Tzedakah to poor people prior to praying at the grave of the Tzadik (In fact it is best to give Tzedakah prior to praying anytime, as the Pasuk says "Ani B’Tzedek Echezeh Panecha- with the Zechus of Tzedaka will I approach Hashem [to pray]. (see Bava Basra 10b)
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Our holy sages, the Chazal tell us
תנא דבי אליהו כל השונה הלכות בכל יום מובטח לו שהוא בן עולם הבא, שנאמר הליכות עולם לו, אל תקרי הליכות אלא הלכות - one who studies [at least two] Halachos daily is assured a portion in Olam Haba - the world to come. (:מגילה כ"ח )