You can save a few frames by dashing from the doorway before stopping to walk diagonally (or pumping) up the stairs. Using the hookshot will speed things up even more as it will allow you to skip the slow stair movement.

The order in which you open the chests (right first vs. left first) is mostly up to preference, but the right path is shown here as Link is able to open the chest even when flush against the right edge of the stairs (not true on the left edge).

While it's theoretically possible to use a hookspeed to travel to the second chest, this is not recommended as the time difference is negligible.

Using the hookshot at all is arguably not worth it for the north two chests, as it saves minimal time and you typically want to have the mirror pre-selected so that you can use it as soon as possible after finding the big key.

If not mirroring back to the lobby, the fastest way to exit the room after opening all three chests is to activate hookspeed on the stairs. If that's not available, dashing is still about a half second faster than walking.