Create a SECOND MX record (ideally with a lower priority) and set this with the new IP address.
What will happen is emails will try the first mail server which will work until you move to the new service, then the emails will try the first mail server, it will fail, so it will try to deliver to the second MX record, which will succeed. Once you have moved the server, delete the ORIGINAL MX record.
As for OWA, just setup a new 'A' record such as tempowa.domain.com and assign this the IP address that will be used for OWA access.
Shaun