My opponent casts Elvish Lyrist and passes turn. I cast Sulfuric Vortex and pass turn. During his upkeep, the soonest he can use the Lyrist to destroy the Vortex is after the Vortex's ability has already gone on the stack, right?
404.2. Triggered abilities aren’t played. Instead, a triggered ability automatically “triggers” each time its trigger event occurs. Once an ability has triggered, it goes on the stack the next time a player would receive priority.
So the Vortex triggers, when he gets priority it goes on the stack, and regardless of whether he responds to it with Lyrist or lets it resolve, he will take the 2 damage anyway.