If you are set for NAT, it should handle that. I am not sure why it didn't. Did you change the IP and DNS while the vm was running.
In Mint, that is really a soft link. I think that it is loaded from /etc/resolvconf/resolv.conf.d/tail. It seems like the dhcp operation via NAT should load that from the host.
If I were going to force a DNS server, I would just use google (8.8.8.8). In fact, I would try using a static IP address even though you are using NAT (just use a high number). Ubuntu has a GUI for setting network settings.
Lou