It is customary to recite Birkat Kohanim during the repetition of the Amida at Minha on fast days, but only within a half-hour of sunset. Although the Hazon Ish (Rav Avraham Yeshaya Karelitz, 1878-1953) was of the opinion that Birkat Kohanim is recited at Minha on fast days regardless of when the congregation prays, the accepted Halacha does not follow this view. Others maintained that Birkat Kohanim is recited as long as the congregation prays after the point of Pelag Ha’minha, but our custom is to recite Birkat Kohanim at Minha on fast days only within a half-hour of sunset. This is the ruling of the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) and of Hacham Ovadia Yosef.
Therefore, synagogues should ensure to schedule Minha for 40 minutes or so before sunset on fast days, so that Birkat Kohanim will be recited at the proper time.
Whenever Birkat Kohanim is to be recited, if no Kohanim are present the Hazan recites "Elokenu V’Elokeh Abotenu" in place of Birkat Kohanim. Thus, if a congregation is praying Minha within a half-hour of sunset on a fast day, and no Kohanim are present in the synagogue, the Hazan recites "Elokenu V’Elokeh Abotenu." If the congregation is praying earlier, then the Hazan does not recite "Elokenu V’Elokeh Abotenu," and instead proceeds directly from "U’l’shimcha Na’eh Le’hodot" to "Sim Shalom."
It is customary to wear Tallit and Tefillin at Minha on fast days. However, many Poskim – including the Rashash, the Hida, the Petah Ha’debir, the Shelah, and Rav Haim Palachi – ruled that when a fast day falls on Ereb Shabbat – as Asarah Be’Tebet occasionally does – Tefillin should not be worn at Minha. According to these scholars, Kabbalistic tradition does not deem Friday afternoon an appropriate time for wearing Tefillin, and thus even on fast days Tefillin should not be worn. Rav Haim Ha’kohen of Aram Soba, who was a student of the great Kabbalist Rav Haim Vital (the most prominent disciple of the Arizal), wrote in his work Mekor Haim that Kabbalistic tradition does, in fact, allow for wearing Tefillin on Ereb Shabbat. However, he wrote this only as a hypothesis, and not a definitive Halachic ruling. Therefore, in light of the fact that numerous Poskim ruled that Tefillin should not be worn at Minha on Ereb Shabbat on a fast day, we should follow this view. Nevertheless, it is customary to wear Tallit at Minha even in such a case, for a number of reasons. Besides the fact that the Kohanim are in any event wearing a Tallit to recite Birkat Kohanim, it is also advisable to wear a Tallit in order to recite the Beracha. There is a requirement to recite 100 Berachot each day, which can be difficult on fast days when we are not reciting Berachot before and after eating. Wearing a Tallit at Minha affords us the opportunity to recite another Beracha, and thus it is customary to do so even when a fast falls on Friday and Tefllin are not worn.
Summary: It is customary to recite Birkat Kohanim at Minha on a fast day, but only within a half-hour of sundown. Our custom is to wear Tallit and Tefillin at Minha on a fast day, except when it falls on Friday, in which case the Tallit is worn without Tefillin.
