I don't fully understand your question. If I have registered myispserver.com, and the host name of the server I named as solo.myispserver.com, solo.myispserver.com is the Fully Qualified Domain Name (FDQN) of the server.
Am I missing something? Can you provide more detail?
If you are just trying to see if your system is accessible, that should be relatively simple to test. Also there is an ISPConfig 3 monitor for your phone that you can install and check the status of your server. This would also tell you whether it's accessible externally.
Look at the monitor app here: http://www.ispconfig.org/download/
Don't forget to go to your server System --> Main Config --> Misc and set the Monitor keyword.