The Spider Boss can be really annoying due to your low health at that point in the game.
Save the heavy attack (
) instead of using it right away after he drops from the ceiling. I had trouble beating the boss until I used the heavy attack after he drops the eggs with the baby spiders. Your normal attacks do enough damage and his attacks are easy to avoid during the first phase.
Also jump behind him when he runs across the ceiling and not under or in front of him. That way you avoid getting hit.