When avoiding the cannonballs, a common strategy is to diagonal all the way to the railings on either side, but some movement can be saved by not going all the way to either side. In either case, pumping can be used for a minor increase in movement.
With the boots, a quickhop can be used to speed up the initial leap off of the ledge. Because this is a north-facing ledge, Link must be facing left or right to quickhop off of it; the most efficient way to do so in this screen is to walk to the ledge, then turn left by pressing cardinal left for at least one frame, followed by a diagonal up-left input to start the quickhop.
One possible strategy for this room is to dash blindly up the corridor. This can sometimes save time, but is highly RNG dependent and generally not recommended. When dashing into a canonball, Link will sometimes be knocked upward instead of downward, but this happens less than 50% of the time.
The above comparisons show a sample time save and time loss with one possible canonball pattern, but note that most of the benefits of the blind dashing are negated if the first canonball happens to be fired on the left side.
A capedash or byrnadash can be used to dash through the canonballs while invincible. In both cases the magic usage will be roughly ~28 magic points (slightly more than 1/5th of a bar). This magic cost equates to a potential end-dungeon cutscene refill time of 28 frames (0.46 seconds), or 14 frames (0.23 seconds) with half-magic, so in all cases time is still saved, assuming the magic is not otherwise needed.
It's always faster to take the left set of stairs, even if starting from the right side of the corridor. If a cannonball is blocking your way, it's recommended to simply wait 10-15 frames (0.16-0.25 seconds) for it to pass and still take the left path.
Without the boots, it's over a second faster to jump down and walk back through the cannonballs instead of going left across the bridge.
With the boots, however, it's slightly faster to dash back across the bridge the way you came.
After collecting the chest, a capedash or byrnadash can (again) be used to dash up through the cannonballs and save time over having to dash back across the bridge. If you're also using the cape for the initial dash through the cannonballs, be sure to reselect the cape in advance of this second dash in order to reset the cape's internal cooldown timer. A "poof walk" should be used to avoid pausing while deactivating the cape.
Like the initial dash, this will cost roughly ~28 magic points, which is a potential end-dungeon cutscene refill time of ~0.46 seconds without half-magic. The cape strategy is thus always time-positive, but the byrnadash may not be worth it depending on the situation.
Without boots, the bridge on the upper level is an opportune location to leverage wallpumping. You can double-pump on the sections with rails on both the top and bottom.