I have a client with a domain and email at GoDaddy. I am moving her to Hostgator for hosting only. She insists on keeping her email with GoDaddy. In short, I need to host on Hostgator and keep email with GoDaddy.
She bought the email service Office 365 Email 1 year from godaddy
You can configure the domain and services like you described. If the domain and email are both with GoDaddy, the easiest way to set this up would be to update the DNS to point to Hostgator's hosting. This is generally done by updating the '@' A Record to point to the IP address of the new hosting plan.
You shouldn't need to make any other changes to keep the email active.