The Ancient Rito Song
You will need to have calmed Divine Beast Vah Medoh before you can do this Shrine Quest. Learn more about it here.
Talk to Bedoli, a female Rito NPC at Rito Village. She will share with you half of a song and tell you that her sister, Laissa, knows the second half.
Go to the general store. Laissa will give you the second half if you can recite the first half. Here is the first half so you can answer her properly:
"The pride of the Rito, pillar in the sky. Its heart lights up, when the sun is high."
Answer correctly and Laissa will give you the second half of the song. Now, you will need some sort of way to create fire. The easiest way is to grab yourself a torch, wood, and a fire weapon or flint. Or, a fire arrow works too.
Go to the Akh Va'quot Shrine near the Rito Village and look south. You will see a lone orange platform. You will need to place fire on top of the platform around 12:30 - 12:35 PM.
You can accomplish this through methods like holding up a lit torch while standing on the platform. Or you could shoot a fire arrow onto the platform. Whatever way you choose, fire has to be on top of the platform at that narrow window of time. We preferred the torch method.
Once the correct time hits, the platform will change from orange to blue, and the Bareeda Naag Shrine will rise up nearby. At this point, the Shrine Quest is complete.