It is proper for men to immerse in a Mikveh on Ereb Rosh Hashanah in honor of the holiday. This should preferably be done after Hasot (midday as defined by Halacha), which generally occurs at approximately 1pm, but if a man wants to immerse during the morning hours, this is also acceptable.
When Rosh Hashanah begins on Wednesday night (as it does this year), such that Rosh Hashanah leads directly into Shabbat, one must remember to prepare an Erub Tabshilin before the onset of Yom Tob on Ereb Rosh Hashanah. He sets aside a piece of bread and a boiled egg, recites the Beracha of “…Asher Kideshanu Be’misvotav Ve’sivanu Al Misvat Erub,” and then recites the text of “Be’den Eruba Yeheh Shari” that is printed in the Mahzorim. The bread and egg should then be put aside in a safe place. It is customary and proper to save the Erub Tabshilin until Shabbat afternoon and eat it with Se’uda Shelishit.
Women light candles before sundown on Ereb Rosh Hashanah at the time printed in the calendars. They recite the Beracha of “…Asher Kideshanu Be’misvotav Ve’sivanu Le’hadlik Ner Shel Yom Tob.” The women light candles on the second night of Yom Tob, as well, after dark, when the men return home from Arbit, before the Yom Tob meal. And, of course, when Rosh Hashanah falls on Thursday and Friday, the women also light candles on Friday afternoon as usual.
It must be emphasized that Halacha forbids creating a flame on Yom Tob, such as by striking a match, even though it is permissible on Yom Tob to light a candle from an existing flame. Therefore, one must ensure before Yom Tob to light a large candle that will burn until the time for lighting on the first day of Yom Tob, and in the case when Rosh Hashanah falls on Thursday and Friday, he must also have a candle burning Friday afternoon so the Shabbat candles can be lit. One should obtain a long-lasting candle before Rosh Hashanah so that the Yom Tob and Shabbat candles can be lit as required.
Furthermore, Halacha forbids extinguishing a flame on Yom Tob, and therefore after a woman lights the candles, she must not extinguish the candle with which she lit, but should rather put it down and let it be extinguished by itself. This Halacha is also relevant for those who smoke on Yom Tob; they may not put out the cigarette, and must rather put it down and let it burn out by itself. (Of course, one should not smoke anytime; but if one does smoke, he must ensure not to actively extinguish the cigarette.)
If a mourner is observing Shiba, Heaven forbid, the onset of Rosh Hashanah ends the Shiba observance. Even if the mourner did not complete seven days of Shiba, the mourning period ends with the onset of Rosh Hashanah, and he gets up from Shiba on Ereb Rosh Hashanah. However, Maran ruled that in such a case, the mourner may not bathe with hot water on Ereb Yom Tob. Although bathing is allowed immediately after Shiba, it would not be permitted on Ereb Rosh Hashanah in this case, as one would thereby be actively “breaking” the Shiba. This is the ruling accepted by Hacham Ovadia Yosef. Therefore, if a mourner is observing Shiba before Rosh Hashanah, he gets up on Ereb Rosh Hashanah but should not take a hot shower until nighttime, after Rosh Hashanah begins, at which point the Shiba restrictions no longer apply. Bathing with hot water is permissible on Yom Tob, and therefore in such a case one should shower on the night of Rosh Hashanah.
Summary: It is proper for men to immerse in a Mikveh on Ereb Rosh Hashanah, preferably in the afternoon. When Rosh Hashanah begins on Wednesday night, one must prepare an Erub Tabshilin before the onset of Yom Tob, by setting aside a piece of bread and a hard-boiled egg, which should be eaten at Se’uda Shelishit on Shabbat. One must also ensure to light a long-lasting flame before Yom Tob so that the Yom Tob candles can be lit on the afternoon of the first day of Rosh Hashanah, and so that the Shabbat candles can be lit on the second day if Rosh Hashanah falls on Thursday and Friday. A mourner, Heaven forbid, ends his Shiba observance with the onset of Rosh Hashanah. He should not take a hot shower on Ereb Rosh Hashanah, and should instead shower at night, after Yom Tob begins.