Running an AD&D adventure ("The Ghost of Harrow Hill") in 5e; any advice?

[BisectedBrioche]

Is the wizard able to cast spells while in ghostly form, but the final boss is him being cut from his magic by being forced into his corpse?

Implicitly; he only has a very basic fear spell on his ghost statblock, but he shows up throughout the adventure to do scary house things™ (like summon enemies or seal up houses).

It's possible that spellcasting undead weren't level appropriate back in AD&D.

That would likely be for the best. Common and uncommon magic weapons can be pretty great and flavorful, and a +2 or +3 weapon in 5e are very significant boosts.

You can also remove some of the weapons, on top of that.

I think what I might do is just roll a random treasure hoard and distribute that (aside from one specific case of a Luck Stone that feels very thematically appropriate). Should be way easier than trying to make a judgement call on how many of the items are too many.

I've done it with Dark Clouds Gather. You have a ton of work ahead of you. You'll need to rework every encounter to ensure it's the appropriate challenge based on 5e assumptions and PC abilities, you'll need to swap creatures that did not make it all the way to 5e, you'll need to tweak BBEG powers (you already know that), and I'm sure there are a few other things I did that I don't remember now.

It was worth it, but damn it was a lot of pre-work.

Aside from the final boss (and some grabbing hand things that I'll probably just make a complex trap), they're all creatures that made it (undead are persistent like that ;) ).