AWS does indeed do static NAT.
That said, as long as you have a virtual host listening for the host/domainname in the certificate, shouldn't configuration be pretty straightforward? What issue are you running into?
Also, are you using an Elastic IP? If not, be warned that your IP will change when the instance is put through a stop/start cycle (reboots take place on the same underlying hardware, but a full stop/start will migrate to new underlying hardware, so you should use an EIP for any DNS-dependent IPs).