You have to start on a new save file and play all the way up to 9-5 and beat 9-5. You may not use Peach, Toad, or Rosalina. You cannot use the following powerups/headgear: Catsuit, double cherry, light box, cannonball box. No multiplayer allowed. Timer ends when you touch the flag on 9-5.
I don't think sub 1:20 is even close to being possible. This is because there are only few levels where Toad's crouch boosts are faster than Luigi's. And even on those few levels, it's only like a second or less faster in ILs. So in RTA, with many starts and stops when getting stars, Toad's crouch boosts would be even less faster. It's also not the crouch boosts making Toad a bit faster, it's that since he jumps lower, he can do more within a section (ex. 5 crouch boosts with Luigi is faster than 5 crouch boosts with Toad. But crouch boosting the entire 5-4 land would be about 1 igt faster with Toad than Luigi as he can do more crouch boosts within that section than Luigi can.) So as of now, it looks like only a 1:28:XX is possible theoretically for now. And with the few seconds of timesave with Toad crouch boosts, it will still be around a 1:28:XX.
The timer stops immediately when you touch the flagpole