You don't need a NAT. You would need a NAT only if you had only one server in the public subnet, and the other servers behind the private subnet. As you have already one server serving websites, I can assume you have a VPC with either public subnet or public subnet + private subnet.
For more info, see http://docs.aws.amazon.com/AmazonVPC/latest/UserGuide/VPC_NAT_Instance.html
Anyway, just launch your linux instance and make sure to configure your HTTPD (Apache VirtualHost or Nginx location) to respond to requests addressed to your domain subfolder correctly.