I'm guessing that everything they do, they do to please their dragon overlords.
So it will all come down to a few simple things:
- Train in combat
- Ensuring the above tasks can continue to be performed indefinitely
Pretty self-explanatory. They would likely have a place to gather and worship together, to honor their venerable deities. They may also have private worship chambers, that would have thick stone so that they can worship and/or meditate in silence.
I don't think there is much needed to ensure this task can keep going. So long as they are alive (sort of), they can worship.
If you're undead and going to live forever, you need someway to occupy your time. They will likely read a lot of material about how and why their gods are divine, and then re-read it, and then keep on reading it until they can dictate every word of it from memory. And then read it some more. This means that they will need some sort of library to store their books, which will likely be the largest room in the monastery. Also the head priest/mage will need to stay quite mentally sharp, so they might have books about casting magic too.
Books will eventually be worn out, so they will need to be copied onto new parchment/paper. This means they will need knowledge of how to make paper/parchment, as well as how to bind them into a book, as well as materials to write such as quills and ink. This supply may come from a small farm (containing whatever birds quills are made from, and goats to use their skin as parchment), or a garden with plants to make into ink (or however old ink was made) and trees to make paper. This means they will need to be proficient with gardening and shepherding etc.
This needs to be done, as if their dragon gods eventually visit them (that's the dream!) they will want the place to be presentable. And I'm guessing the dead produce a lot of dust, as they will consist entirely of dead cells. They will also need to make sure any of their tools for performing tasks (brooms, book binding equipment etc.) are repaired or replaced.
This means they will need some sort of workshop to repair/make tools. This will likely be confined to its own area, such as an external shed or a side room that is cut off from elsewhere, as the noise and waste such as sawdust will pervade the common areas for worship and reading. All their common areas will also likely be quite bare, in order to make their cleaning easier, and I can't imagine they're materialistic or sentimental enough to keep things around for decoration, they would only own practical things like tools and shelves (and possibly chairs?).
Train In Combat
There's always the possibility that others might try to invade their monastery, so they will need to be able to fight them off. This means they will need to stay honed on their fighting skills, and also they will need weapons in good condition. Again, they might be visited by the dragons who will want them to march off to pillage at a moment's, so they will need to be prepared for that eventuality.
This will require an armory, training grounds, and a smithy to make weapons. They will likely need to find the materials for weapons themselves, so perhaps also a mine of some sort to find iron.
That is everything that I can possibly think of that they might need rooms for, or have inside their home.