Narset, Parter of Veils interaction with Aria of Flame
You have it correct, they will be allowed to draw cards.

The specific order is:


Opponent casts Faithless Looting... that casting process looks completely normal and doesn't interact with Narset at all.
Aria of Flame's ability triggers, going on the stack on top of Faithless Looting
Aria of Flame's ability resolves, dealing damage to Narset
Narset goes to the graveyard due to having 0 loyalty counters
Faithless Looting resolves
Faithless Looting acts completely normally, because Narset is not around.