So I want to complete the Temple of the Moon quest that Simon gives you in the caverns of the Academy, but when I died in the lower dungeons when I was still at a lower level, I lost the jug he gave me. Is there any way to recover the jug or get a new one?

None

You will definitely have the opportunity to do the quest with another character.

But, probably the most important thing to make sure you know from this:
You should visit your corpse and pick up your belongings when you die.
If you die in a place you can't reach or find as a ghost, your belongings will show up in your pit (whic is at the place you recall to). So, if you die and can't find your corpse, you can recover all the belongings that aren't picked up by someone else by waiting by your pit until they arrive inside the pit.

When people delete, or Lowbie dies and gets looted, or dies, and their stuff decays to the pit...

You've got until your 6th rank, since you can't get into the Academy after that. If you run around to the pits, and then do other things you might just stumble upon another one to use.

-Cassman-

I'm guessing you could get the jug from another character. You would have to convice them to give up on the quest and give you their jug but that might be fun to try and do!

Just to note I've never done this so have no idea if it would actually work.

That said, this quest and all quests are completely optional, and there's quite a bit to do here. Just move on and don't worry too much about this one.

valguarnera@carrionfields.com

At least I don't know how it could be possible besides finding the jug in question.