Take Backup of your CRM files and on the current setup use the Upgrade Wizard in Admin and run the Update from 7.13.0 to 7.14. Once that is done, update PHP version and run crm files.
After updating php, do you menan that i should unpack the original suitecrm backup of all the files to the installation folder and that way replace the ones, some of which which maybe have changed during upgrade?
You should be fine if you run the upgrade using the PHP version that fits the old version of SuiteCRM, and then right after the upgrade finishes, even before running a QR&R, swap PHP version to the one that fits the new SuiteCRM version.
Make full backups before, to ensure you have a way of restarting the process, if necessary.