I lost my right airpod, can I synch my old right airpod to my new case?

I doubt it, but you can try. As I understand it, the serial numbers on both AirPods and case must match for proper operation. That is why Apple needs to know the serial number for a replacement. Seer the 'Replace a missing AirPod or case' section in:

How to find your lost AirPods, AirPods Pro, or AirPods Max - Apple Support

and you have to go through this replacement procedure:

Set up your replacement AirPods or charging case - Apple Support