The IP address really has nothing to do with your certificate. The only time an IP address is important is if you are not using SNI on your server to allow you to share multiple SSL certificate on one IP address.
This process is really very easy. If you look in your Apache configuration for the three lines that contain your Private Key, Intermediate Chain Certificate and your issued certificate you can simply copy those three files to your other machine and use the same configuration settings in your Apache configuration on the new machine.
Without knowing more details about your Apache configuration it will be hard to be more specific than that really. Feel free to elaborate if necessary.