Where to Start?

Alright, so me and a friend decided to challenge ourselves to see who could hack into the other's computer faster. Whoever manages to (remotely) create a text document in the desktop wins. What I didn't forsee is that he would be using a dynamic dns service (possibly no-ip). I can't think of any way to find out his url or private IP, and couldn't really find any answers by searching the internet. Furthermore, social engineering is unlikely to work, since we both know that we are trying to get into each others's PC and will be cautious. So my question is, how can you conduct reconnaissance on a pc not on your same network, who is using a ddns service, without even knowing the IP or url, if doing that is even possible? (Sorry for any grammar mistakes, English isn't my first language :) )