Throughout the year, the kohanim (“priests”) do not lift their hands [to bless the people] during Mincha services, because people [usually] eat a meal before Mincha and we are concerned that the kohanim might bless the people when they are drunk, which is forbidden. On fast days that have a Ne’ilah service, like Yom Kippur and the fasts [that the Rabbis instituted] for droughts, the kohanim bless the people [during Ne’ilah], because there is no reason to fear that they will be drunk, seeing that it is a fast day. During Mincha of those days, however, the kohanim do not bless the people for fear that they may mistakenly think that they are supposed to do so on regular days, as well. Regarding ordinary fast days, on which we do not pray Ne’ilah, [the law depends on when the congregants pray Mincha]. If they pray at the same time that Ne’ilah is usually said [i.e., shortly before sunset], the kohanim bless the people (cf. Ta’anit 26b; SA, OC 129:1). But if the congregation prays Mincha earlier, Birkat Kohanim is omitted, since it is not the time designated for Ne’ilah. In such a case, the cantor, as well, omits “Elokeinu v’Elokai Avoteinu,” which is customarily said when no kohanim are present.
Therefore, it is fitting to call Mincha on fast days for a time that enables people to merit [participating] in the mitzvah of Birkat Kohanim. Ideally, one should pray Mincha within half an hour of sunset, which is the best time to pray Ne’ilah. Nevertheless, as long as the congregation prays after plag mincha, the kohanim may lift their hands [and bless the people]. If they pray earlier than that, however, Birkat Kohanim is omitted.
A kohen who is not fasting should not ascend the platform [to bless the people]. And if there are no other kohanim, some authorities say that he [still] may not go up (KHC 129:5, Torat HaMo’adim 3:4), while others maintain that he should. [The latter opinion] goes as far as to say that he should go up even if there is one other kohen (Lu’ach Eretz Yisrael; Halichot Shlomo, Tefilla 10:13). If there are less than six people fasting, no kohen should go up to bless the congregation during Mincha, even if he is fasting (see Piskei Teshuvot 129:2).
If the cantor’s repetition of the Shemoneh Esrei continues until after sunset, the kohanim are permitted, be-di’avad, to lift their hands [and bless the people] up until tzait ha-kochavim, for there is a combination of uncertainties here: 1) According to Ra’avyah, Sefer Yerayim, and Or Zaru’a, kohanim are allowed to bless the people at night. 2) Twilight is possibly still daytime. 3) Rabbeinu Tam holds that the period after sunset is definitely daytime, and [the following poskim agree with him]: Shulchan Aruch HaRav (623:8); Piskei Teshuvot (623:13), quoting Rav Shlomo Zalman Auerbach and Rav Elyashiv. The authors of Yechaveh Da’at (6:40) and Or Le-Tziyon (vol. 2, 8:13) concur, adding that [this period lasts] 13.5 minutes after sunset.