It would be best if you could open up your firewall for all of http://www.linomasoftware.com/ (the port is default 80)
If this does not suffice, you can specify just the upgrade XML file (The file that the Check For Updates reads to determine whether or not an upgrade is available). It is located at http://www.linomasoftware.com/upgrader/goanywhere/upgrade.xml. Assuming you can access this file the Check For Updates page will show you instructions on how to upgrade and will give you an option to download the files. If you only allow access to the XML file then you will need to download the upgrade file to your local PC and manually move it to the system that GoAnywhere is running on because the server will not have access to the upgrade files directly.
If opening up the ports on your firewall is not an option you should go to the download page at http://www.goanywhere.com/download, fill out the information, and download the appropriate files manually.