Fighting the Ancient Wyvern on the ground is not only dangerous, but utterly pointless when the battle arena is designed to do otherwise.
Ignore the Ancient Wyvern when it swoops in and rush up the stairs, turning left to enter a building with some Serpent-men. Roll past any attacks as you exit the building to the left and then turn right as soon as you enter the next building to find another staircase.
Use an Estus Flask if you’ve taken any damage and then head up the stairs and ignore more Serpent-men and continue across the bridge. Continue until you spot a fog wall where the boss area ends and climb the ladder to the left, taking cover around the corner if the Ancient Wyvern is breathing fire at you, and then drop onto another platform on the other side.
From here you’re safe from the Wyvern’s flame breath attack that arcs into the air and you can freely use ranged attacks until it dies.
Drops: 70,000 Souls , Dragon Head Stone