In most cases, yes. Do a module upgrade on the v13 machine to the best of your ability to get the most current v13 modules. Then run the FreePBX backup. Transfer the backup to a local PC for storage.
When you spin up the v15 machine, the backup should restore. The CDR data was a problem due to changes in the database and I believe that was addressed in the v15 backup restore function. On the v15 backup restore page after uploading the legacy backup, there is a button that says something like “Restore withLegacy CDR.” That should get the CDR data imported.
In any case, I’d put the v15 on a new machine or in the cloud and try the restore before tearing down the old v13 machine. I would not recommend doing an in-place upgrade under any circumstance.