You should set up what is called a backup MX relay to accept mail while you do the switch over. This is a mail system that typically has a lower priority than your primary mail server and is just a queue for mail. It will accept mail if your normal mail server is down. You should also shorten the TTL in advance in your DNS records so that when you do the DNS switch it propagates quicker. When the short TTL DNS records have propagated, switch the old email server's IP address for the new email server's IP address (as you don't want new mail going to the old server).
When a mail server tries to deliver mail to a domain, the servers in the MX field with the lowest priority number are tried first. If there are two MX servers listed, and they have the same number for their priority, then they are connected to in a round robin fashion until one of them works.
If you don't want to set a backup MX yourself, you can ask your hosting provider (if you have one) for one instead. My hosting service charges a small nominal fee for me to use their backup MX relays.
I have a 'primary' MX which is my mail server. Their servers will receive my mail instead if it fails and attempt to deliver it to the primary at regular intervals. The results of
dig MX look like this:
;; ANSWER SECTION:
mydomain.co.uk. 86400 IN MX 10 mydomain.co.uk.
mydomain.co.uk. 86400 IN MX 20 mx1.mythic-beasts.com.
mydomain.co.uk. 86400 IN MX 20 mx2.mythic-beasts.com.