Before I suggest anything please note that restoringRestoring a snapshot simply creates a new RDS instance. So if you restore the prod DB snapshot to staging then you will be creating a new RDS instance with the restored data in it.
-If both of the RDS instances are in the same aws account then You will either have to delete/rename the staging DB and then restore the prod DB
-If both of the RDS instances are in different/separate AWS account then you can share the snapshot in other account and restore it there Here is the aws official doc : https://aws.amazon.com/blogs/aws/amazon-rds-update-cross-account-snapshot-sharing/
If both of the RDS instances are in the same aws account then You will either have to delete/rename the staging DB and then restore the prod DB
If both of the RDS instances are in different/separate AWS account then you can share the snapshot in other account and restore it there Here is the aws official doc : https://aws.amazon.com/blogs/aws/amazon-rds-update-cross-account-snapshot-sharing/
Ps.Another: Another option can be, using mysqldump to take backup from prod and then restoring it to the staging instance