Zechariah's division was on duty … he was chosen by lot: It was common in Old Testament times to cast lots (similar to throwing dice) to determine God’s will (1 Chronicles 26:13–16; Nehemiah 11:1; Proverbs 16:33; Jonah 1:7). There were 24 orders of priests that took turns serving in the temple (1 Chronicles 24:1–19). They cast lots to choose which priest would serve in the sanctuary of the temple. There were many priests, so this honor might come only once in a priest's life. The angel appeared at the most holy moment of Zechariah's life.
to enter the temple of the Lord and burn incense: Luke uses the Greek word naos to refer to the temple building. He did mean the whole temple complex (hieron), which he refers to as "the temple courts" (Luke 2:27). The temple complex included the Holy Place and the Most Holy Place (see Hebrews 9:1–5). The priests burned incense twice a day in the Holy Place.