Note: springrts-0.82.7 is outdated. You will be able to play single player but not online without version 84.0.
Part of the problem with Linux instructions are that there are so many distributions out there. I am all for recommending some suggestions for better instructions if they would ease the installation process. Fundamentally it is:
-Install latest spring engine (84.0 now)
-Install springlobby (0.139 or 0.139-X)
-Install the zero-k mod (few options for this, rapid is recommended)
-Make sure you can get the maps needed (springlobby should download the map for a battle when you try to join it)