Question says it all. I have a digital ocean IP address and I want to know if I can use bind9 to resolve a registered domain on a completely different network/IP? If not, what can I use?
Yes sure you can.
for your case you have just to insert the external IP in forwarders in the /etc/bind/named.conf.options file
Please read this to learn the full configuration of bind9 in Ubuntu. How do I do a complete BIND9 DNS Server Configuration with a hostname?