Should this be a corporate network like dmorgan suggests, you might have other resources for home users, like VPN...

Ask your system administrator if you can get a VPN account and the necessary software. With a VPN connection you should be able to get onto the machine with the local IP.

And I'm sure someone has configured your router... he or she could set up port forwarding for you... Which port to use depends on the application... 3389 for example is used by Microsoft's RDP client.