In order to change your MX and A records, you'll need to use our DNS settings, explained in this guide: https://www.digitalocean.com/community/tutorials/how-to-set-up-a-host-name-with-digitalocean

This may take a short time to propagate, depending on the time to live (TTL) settings for your domain. If your previous droplet is still running at the older address that the MX and A record is pointing to, you should experience minimal downtime, as your DNS settings will be going from one active server to another.