When the ancient one awakens, do we pass the first player token?
The rules for The Final Battle (page 22 of the rulebook) state that the first player token should be passed at the start of the turn, when the investigators refresh:

1. Investigators Refresh
At the beginning of a round of combat with an Ancient One, the investigators may refresh their cards, use any character abilities, and adjust their skill sliders as though it were the Upkeep Phase. Next, the first player marker should be passed to the left. Finally, they may trade items among themselves as if they were in the same location (see “Trading Equipment” later in these rules).

(Emphasis mine)