First, there's a couple terms here. A192.168.x.x address is a private IP address as you have stated. Most likely, it is also dynamic in that your home router is probably handing out IP addresses as computers ask for them. It does this using a built-in DHCP server that serves just that purpose. 192.168.1.1 is the gateway address your computer uses to reach servers on the public Internet. It is also the address of the LAN (home network) side of your router.
Your router also has a public IP address of 81.97.11.231 on the WAN port. The router has the job of moving data to and from the private LAN to the public WAN. However, it's WAN address is probably also dynamic, not static. That is, your router probably asks your ISP to assign it an IP address by sending a request to your ISP's DHCP server. If you were to shut off your router for a few minutes and turn it back on, it's likely to get a new, public IP address that's slightly different than your old address.
All that said, you don't need to turn your computer's IP address into your router's public IP address. Instead, you need to add something to your router so that it knows when certain requests are received (such as the GameSpy Arcade requests), it needs to forward those requests on to your desktop. That "something" is called port forwarding. http://portforward.com/ has a lot of useful information about the topic.
In order for GameSpy to try to communicate directly with your PC, it needs an IP address and a port number (from 0 to 65535). Certain port numbers are reserved for very specific services like reading email (port 110 for POP3) or web servers (port 80). Similarly, GameSpy uses a couple ports to talk to your desktop. You need to let your router know that you wish these requests to be forwarded to your desktop.
Basically, a port forward is a rule that tells the router that if it gets a certain type of request (either TCP or UDP in this case) to a particular port, you want it redirected to a particular machine on your home LAN. For GameSpy for example, I forward all UDP requests to port 13139 and TCP requests to port 28910 to my desktop's IP address (e.g., 192.168.1.100). Exactly how this is configured varies by routers, but look for an "advanced" menu or the term "port forwarding"/"port triggering" or "gaming" or something similar. You should be able to type in the port or a range of ports, the protocol (UDP or TCP), and the target machine on your LAN (i.e., 192.168.1.100) to which such request should be sent.
Since you don't want to have to redo these rules every time the IP address of your desktop changes (Because it's dynamic, that could be at any time), you probably want to change the IP address of your desktop to being a static IP address. Find out the range of IP addresses your router's DHCP server serves (by looking at your router's configuration pages). Let's say you find out that's 192.168.1.100 to 192.168.1.149. You will want to give your desktop a static IP address outside that range. (e.g., something like 192.168.1.50)
First, open a command prompt window, type "ipconfig /all" into that window and note all the numbers and settings for your current IP configuration. You will reuse the
Subnet Mask: 255.255.255.0, Default Gateway:192.168.1.1 and DNS Servers info.
For Windows XP, open the Internet Protocol (TCP/IP) Properties dialog of your Local Area Connection Properties. Select the "Use the following IP address" and "Use the following DNS server addresses" radio buttons and enter the static IP address you decided upon. The netmask and other you enter will be that you found from the ipconfig /all command.
Finally, you may need to add rules to the Windows firewall on your desktop to allow incoming TCP and UDP requests to the ports you just forwarded. Simple, huh? :)