Hi,
Making a copy of a DNS zone in another DNS server is really easy. Just just have to create an secondary DNS zone on the new server with the same dns suffix and declare IP address of the DNS server that owns the zone (DNS server of DOMAINOLD in your case).
Add to that, don't forget to allow zone transfer with any address on the DOMAINOLD DNS zone on the "old" server.
As soon as the zone transfer is complete (check the content of the DNS zone to see if all the records are copied) you can then change the zone type on the new server and make it become a primary DNS zone.
So, you'll have a copy of the "DOMAINOLD" zone on your new DNS server. But I wonder what you want to do with this DNS zone on your new domain. The new domain won't have the same name I suppose.
If you're planning to finally have a new "DOMAINNEW" DNS zone that contains same records than the old "DOMAINOLD" DNS zone you won't reach this goal by this way.