First download the Game center package here.
1). Open Cydia and install "GC Patcher - iPhone 3g"
2). SSH into your device and go to /private/var/stash/Applications
If you dont know how to do this then follow the tutorial here.
3). Copy Game Center~iphone.app (from the download) to that location. Right click on this folder and set permissions to "0755".
4). Navigate to /system/libary/CoreServices/SpringBoard.app/ and open up the file "N82ap.plist". Add the line "<string>com.apple.gamecenter</string>" under "<string>com.apple.AppStore</string>". Save this file and close it.
5). Navigate to System/Libary/LaunchDeamons/ and copy com.apple.gamed.plist (from the download) into this directory.
6). Lastly navigate to /usr/libexec/ and copy the gamed folder (from the download). Set permissions to "0755".
7). Finally, reboot your device and Game Center should be working perfectly!
Thanks to MrDodi at iPod Touch Forums for this.