It is possible to climb without wasting stamina when jumping at the wall on the right spot (about 25% to the left). Whistle Sprinting near the exit can be heard by the attack bug (avoid whistling near the exit if you need it). To grab the Attack Bug from the intro cutscene, empty your stamina running in it's direction, then crouch jump twice.
After talking to the old man, hold the direction toward Vah Medoh, shield surf down the mini hill, shield surf again on the way down, jump to reach the Guardian in the water, jump from the Guardian to the other side of the lake, then alternate whistle/normal sprinting to Magnesis.
Exit Shrine of Resurection, go left, equip a shield before grabbing mushrooms, that way the mushrooms are the last item you interacted with, allowing fast menuing. Climb up, and go straight to the snow with a shield surf. Use jump to minimise interaction with surfaces that may slow you. Eat a mushroom (not mandatory with optimal speed). Shield jump over the broken bridge.
You can go near the Bokoblin fire camp to reset the cold damage timer to not have to eat. Shieldsurf as much possible. Use Fall Damage Prevention to land near Stasis Shrine.
Stasis the boulder to the left, hit him from the left, un-stasis it, grab the bow from the chest, Stasis Launch in the direction of Bombs (add a bow shot to cover more distance). As soon as Stasis is off cooldown, use a tree to Stasis Launch to Bomb Shrine.
Optional: Stasis launch via the metal door behind the shrine. Else just run to the closes tree. Stasis Launch with the tree to the top of Temple of Time.
Jump off the tower northbound having a two-handed weapon in hand and shieldsurf in the air. Once you got the boost from shieldsurfing, put the shield away and mash the attack button to make link attack in the air.
This will give him enough forward momentum to reach the water and with that prevent fall damage.