From Rosh Hodesh Nissan through 13 Sivan (a week after Shabuot), those who have the custom to recite Tikun Hasot every night omit the Tikun Rahel section, and recite only the section of Tikun Leah. This is mentioned by the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909), in Parashat Vayishlah (Shana Rishona, 5-7). Of course, on Shabbat, Yom Tob and Hol Ha’mo’ed, Tikun Hasot is not recited at all.
The Shulhan Aruch rules that it is permissible to observe a personal fast during the month of Nissan. For example, if one observes a Yahrtzeit for a parent during Nissan, he may follow the custom of fasting on a Yartzheit. This is the ruling also of Hacham Bension Abba Shaul (Israel, 1923-1998), in Or Le’sion (vol. 3, 5:1).
The work Kav Ha’yashar records the custom not to visit cemeteries during the month of Nissan. If one observes a Yahrtzeit during Nissan, then according to this custom he should visit the cemetery on Ereb Rosh Hodesh Nissan, rather than on the day of the Yahrtzeit. Common practice, however, does not follow the custom, and it is customary to visit a cemetery during Nissan if one observes a Yahrtzeit or completes the Shiba or Sheloshim mourning period. Nevertheless, one should not move himself to tears during the visit, given the festive nature of the month. (This Halacha is mentioned in the work Ner Le’sion.) One who fears that he might be moved to tears should visit the cemetery on either Ereb Rosh Hodesh Iyar or the 15th of Iyar.
There is a custom to donate money for Ma’ot Hittim (literally, "wheat fund") during Nissan. In the olden days, community members would donate money to the poor so they could purchase flour with which to bake Masot. Nowadays, the custom is to give money to the needy to help them with their Pesah purchases. The Zohar Hakadosh emphasizes the stringency of this obligation, applying to those who do not participate the prophet’s warning, "I shall cast excrement upon your faces – the excrement of your holidays!" If one enjoys the holiday celebration without ensuring that the needy are also able to celebrate, G-d disapproves of his celebration, and he is liable to grave punishment, Heaven forbid. Similarly, the Rambam (Rabbi Moshe Maimonides, Spain-Egypt, 1135-1204) writes that one is obligated to bring joy to "G-d’s children" (the needy) on holidays, and if one rejoices on Yom Tob without giving to the poor, this is not a celebration of Misva, but rather a "celebration of his stomach." In earlier generations, Bet Din was authorized to force people to contribute to Ma’ot Hittim. Everyone is obligated to contribute according to his means, and those who do not participate in this charitable endeavor commit a grave sin, because this is an ancient custom and the poor rely upon these contributions for their holiday needs.
Summary: Those who recite Tikun Hasot every night omit the Tikun Rahel section from Rosh Hodesh Nissan through the 13th of Sivan. One may observe a personal fast during Nissan, such as for a parent’s Yahrtzeit. One may visit a cemetery during Nissan, though one should ensure not to move himself to tears. There is a strict obligation of "Ma’ot Hittim," which requires donating charity before Pesah to help the poor with their holiday expenses.
