Start trouble shooting this to see if you can access your server from your home LAN. If you can, you know it's a router issue.
The default port for subsonic is 4040. On your router you should do a couple things.
1. make sure your computer running the subsonic server ALWAYS gets the same IP. There are a few ways of doing this, like specifying the MAC address always gets the same IP. Or setting up a static IP for that computer.
2. forward that PORT in and out of your LAN. Check your router brand and model through
http://portforward.com/. They have excellent instructions for setting up a proper port forward.
While at home, you can put your port to check, 4040, into here
http://www.canyouseeme.org/, and it's a good way to see if your 4040 port is forwarded properly.