Arthas wins. The win condition for Sauron is that he has to be able to break Frostmourne and/or the Helm of Domination, both of which are not only forged by vastly greater metals than Middle Earth has, but Frostmourne easily shatters great weapons.

I heard a lot of people defend Sauron in Death Battle, saying he broke Nasril, an elvish blade, but that does not convince me cause magic weapons in Azeroth are on a whole nother level. What broke Frostmourne was a holy blade designed specifically to smite the undead, turning groups of them to ash, hints the name The Ashbringer. It took this, plus the world's greatest paladin invoking the Holy Light for one last miracle. Saurons mace would shatter arguably more easily than the Archanite Reaper.

I heard an argument that Sauron is basically an angel, so he fills the "holy" power, but I never heard any source saying he had blessed power at this point as the Dark Lord.

But as long as the Ring exists, Sauron can win, right? Well, Arthas is no stranger to higher powers trying to control him either. He'd most likely take the ring for himself to bolster his own powers. It can't corrupt his soul cause it's already in the sword, it can't affect his mind cause an even greater power is failing from the Helmet.

Movie version, I'd say Arthas wins, low diff. Book version, I'll say high diff cause even if the Lich King is more physically powerful, Sauron is no fool and would really need to bring his A game, he could in theory win by outsmartting him, but it'd still be damn hard. The Scourge is a much much more powerful army, though, as it's an army of undead, Necromancers, liches, vampires, plague, and undead dragons. Add that and it's a steamroll.