Not sure what's Ulysses, but you could probably set a Github action for it? This way it would be rebuilt and deployed on pushing into your git repository (hopefully it's possible to work with git from an ipad?)
It is possible to work with git from an iPad. I've set up a similar workflow (not for mdBook but for gitbook) a long time ago. I used Working Copy and Textastic.
https://github.com/peaceiris/actions-mdbook