I am running Ubuntu, in theory this fix should work on any Linux distro AFAIK. Download libpng12, I am not allowed to post links so Google "debian download libpng12 i386" gets the page I downloaded it from
Extract the .deb file and copy the lib folder to the games directory (default ~/GOG Games/Zombasite) so you should now have ./Zombasite/lib/i386-linux-gnu which contains library files.
Edit (backup first) the start.sh file add the following above "# Initialization"
export LD_LIBRARY_PATH=/home/user/"GOG Games"/Zombasite/lib/i386-linux-gnu/
Now the game should start
EDIT: As I can post links now, here is a link to it for if anyone else needs it:
https://packages.debian.org/jessie/i386/libpng12-0/download