Instructions
Find a way to reduce your HP to below 30 percent before the fight for the Clever Rat's ring to take effect. Use the Makoto or jump from heights high enough to hurt but not kill you.
Bring approximately 500 arrows to the fight, preferably light or heavy. A Viscous Compound Long Bow is a good choice for the battle.
Shoot the dragon as he breaths fire onto the middle of the bridge from a safe distance. The dragon will fly to another location after you bring his health down to about 50 percent. Continue across the bridge.
Proceed to the dragon's second location. Stay at a safe distance and shoot at the dragon's mouth when he breaths fire. Alternatively, you can find a "blind spot" near a pillar on the right where the dragon breaths fire but cannot hit you. Lock onto the dragon with your bow from the pillar and shoot until he dies.