Hate to assume the worst, but I wouldn't think you would need to mess with ports for this game to work. I personally have not had to mess with router ports for any games in years - only for p2p file transfer software (uTorrent, etc). And since your Laptop can play the game without a problem, then it stands to reason that the necessary port is already open on the router.
Do you have any internet connectivity on this new computer at all? If not, then the problem is not a closed port, it would either be:
A. Your new pc is is not picking up the MAC address (Hardware location) of the router, or vise versa - if this is the case, simply reboot the pc and router simultaneously.
B. It's possible, although unlikely, that your router has the physical eternet ports shut off, in which case you'd have to find out how to turn them back on (See below for how to access admin options in your router)
PS - You could try connecting the PC directly to the modem as well, just to make sure the PC is able to connect. That would also help you to make sure that it is not a closed port in your firewall software on the PC.
http://portforward.com/
This website has guides for port opening and forwarding for most routers. Forwarding is a method of setting all traffic on a specific (or range) port to be sent to a specific device on your internal network. If you cannot find your exact model, look for a similar one and follow the guide as best you can.
Access portals are usually on the same local IP address among all models of a specific brand. (That is the 192.168.x.x). You can get to that by typing in the address in your web browser.
At that point, you would need to login to your router as admin. Common user/pword combos are User: admin / pword: admin, or user: admin / pword [blank].
Then you'll have to find the section that deals with your internal network, usually labelled LAN/WAN, or something similar. You may have a subsection called Port Forwarding. This would be used for just opening ports as well.
From there you should see some sort of port list, and you would just add whichever port as an exclusion to the firewall; you shouldn't need to actually forward it to your PC unless you have multiple PCs with the same port being listened to.