The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) writes that the Arizal (Rav Yishak Luria of Safed, 1534-1572) had the custom to wear the special large Tefillin known as "Shimusha Rabba" when praying Minha on fast days. Most people do not have access to these Tefillin, but those who do should try to wear them during Minha on fast days, in accordance with the practice of the Arizal. The Hid"a (Rav Haim Yosef David Azulai, 1724-1806) writes that those who do not have "Shimusha Rabba" Tefillin should wear Tefillin Rabbenu Tam during Minha on fast days, and this is, in fact, our custom. No Beracha is recited when putting on this Tefillin.
When Minha is recited within 45 minutes of sunset, the Kohanim recite Birkat Kohanim during Minha, even though they already recited the blessing at Shaharit. A Kohen who is not fasting does not recite Birkat Kohanim, and therefore he should leave the synagogue just before the Kohanim recite the blessing. The congregation should try to recite Birkat Kohanim before sunset, but it may be recited even afterward, during the 13.5-minute period after sundown. This is the ruling of Hacham Bension Abba Shaul (Israel, 1924-1998), in his work Or Lesion (vol. 2, 8:13).
Summary: During Minha on fast days, it is customary to wear Tefillin Rabbenu Tam. When Minha is recited within 45 minutes of sundown, Kohanim who are fasting recite Birkat Kohanim at Minha. Birkat Kohanim should preferably be recited before sunset, but if it was not recited before sunset, it may be recited within 13.5 minutes after sunset.
