The past history of Arthur Douglas on Earth-9997 mirrors that of his Earth-616 counterpart. He participated along with other allies of Captain Mar-Vell in the assault and defeat of Thanos powered by the Cosmic Cube.
Since his death he has remained in the Realm of the Dead, fighting pointless battles against the resident super-beings that have also died. After Mar-Vell destroyed Death and created Paradise, Drax was one of the many super-beings that who would be brought to this new afterlife. There he would consume a shard of the Cosmic Cube to create his own ideal paradise.
His fate following the invasion of Paradise by the Kree remains unrevealed.