“Chosen by lot to burn incense”
He was chosen by lot, according to the custom of the priesthood, to enter the temple of the Lord and burn incense. (Luke 1:8–9 BSB)
What it sounds like today
Administrative detail about temple scheduling.
The cultural key
Priests were organized into twenty-four divisions (1 Chronicles 24), each serving in Jerusalem two weeks a year; Zechariah belonged to the eighth, Abijah. With thousands of priests and the lot deciding who offered the incense, a priest might receive this honor once in a lifetime, and many never did. Burning incense at the golden altar, feet from the Holy of Holies while the assembled people prayed outside, was the summit of an ordinary priest's entire career.
Recovered meaning
Luke is telling you this is the biggest day of an old man's life before anything supernatural happens. The angel appears at the exact moment Zechariah's lifelong faithfulness finally reaches the altar; the timing itself is the first grace note of the Gospel.