If you're using the same ip address as used before, there's no reason why it shouldn't work if added correctly. The problem is MSFT has made what should be a pretty simple process unnecessarily complicated. My advice is avoid using MSFT's nameservers; park your domain so you're using your registrar's (or other) nameservers. Add the records needed - you can get the 365 records from admin>>domains>>dns settings.
At minimum you'd need:
A for the web site and CNAME for www
MX for email.
CNAME for autodiscover