His fighting strategy involves teleporting near you, using his sting attack to inflict sickness and dealing some damage along the way. After that, he will usually leave you one turn for you to attack him, and then teleport away. The damage he deals is moderate, but manageable - after teleporting away, he will usually leave you alone for long enough for you to recover from the damage. The main problem is his sting attack that may cause illness, and will kill you if you can't heal from it. Thus, it's easier to defeat Scorpius if you have a way to cure yourself from illness, either via healing potions, spells, or preferably, a unicorn horn. Alternatively, spend some time kicking trees in the upper filler level - each tree will give a bunch of fruits exactly once, and you might find some eucalyptus leaves.
The easiest way to defeat Scorpius though, is to play defensively, by dropping a scroll of scare monster and waiting for him to come after you, and then attack him when he teleports near you. As he lacks any sort of ranged or spellcasting attack, this renders him completely harmless. His pitiful AC then makes it very easy to finish him off.
Another easy way to defeat Scorpius is to use range. Get far enough from the upstair so that you can just hit the upstair with your bow and your original +2 arrows. Wake and wound Scorpius remotely, by arrow, wand, thrown dagger, dart or other means. He will then teleport to the upstair. As long as you stay far enough away, Scorpius will not flee upstairs, and continued fire should easily defeat him.
Since early times, the Scorpion has represented death, darkness,
and evil. Scorpius is the reputed slayer of Orion the Hunter.
[...] The gods put both scorpion and hunter among the stars, but
on opposite sides of the sky so they would never fight again.
As Scorpius rises in the east, Orion sets in the west.