If the server you intend to run your UOX3 shard on is behind a NAT router, then you might have to forward any incoming connections on port 2593 (or whichever other port you decided to use for UOX3) to the internal LAN-IP given to your server by the router. This is called Port Forwarding (wikipedia link), and whether or not it is required depends on the router and how it is setup already. The easiest way to find out is to have someone try to connect to your UOX3 server over the Internet. If they are unable to connect, and you've ruled out all other options (mis-configuration of UOX3/UO Client, firewalls running on host/client computers, etc), then you might have to setup port-forwarding on your router.
How do I do Port Forwarding?
The exact procedure varies from router to router, but in general it involves connecting to the router's IP through your browser/Windows Explorer/Linux equivalent, entering the user-name and password required to log on to the router (these can either be default to the router-brand, or custom depending on your Internet-provider), and then adding the UOX3 port to whatever port-forwarding configuration exists for your router.
Various sites on the World Wide Web include step-by-step instructions for how to setup port-forwarding on different types of routers. One such site is PortForward.com. If the default username/password specific to your router (usually mentioned on these sites) is not working, you might have to consult the documentation that came along with the router, or contact your ISP to get them to provide the username and password.
How can I check if I set it up correctly?
To quickly check if you setup your port-forwarding correctly, you can visit CanYouSeeMe.org (Open Port Check Tool) and just enter the port you're trying to forward. Click the 'check' button, and the tool should tell you whether or not it can see a service running on your server through said port.
The other option is of course to have someone actually try to connect to your shard!
