Assuming your IP address on the Internet is dynamic, a dynamic DNS provider such as no-ip.com will be essential. There's a list of them on Wikipedia
How do you connect to the Internet? If there is a NATting router involved you will need port-forwarding
to send incoming requests arriving at your internet connection to your Java GUI (if it were an HTTP server you would forward port 80).
Regards switching to a server version of the distro, there's not much point -- you could simply install whatever packages are required on your current Ubuntu version, including whatever you choose for the trend and archive data.