You've got multiple problems there. First, looks like you're using CEN64-Qt which won't work with Mupen64Plus. Download Mupen64Plus-Qt
here.
The launcher will be useless though unless you get Mupen64Plus working.
The problem: failed: libpng16.so.16: cannot open shared object file: No such file or directory
This error means the executable you downloaded is linked against a library that is not present on your system (Debian and Ubuntu don't have a package in their repos that contains that version of libpng AFAIK). You could recompile Mupen64Plus to link against the version of libpng on your system, but a much easier way to get things working is to just install Mupen64Plus from your distribution's repos. Assuming you're using Debian or Ubuntu, this should work:
sudo apt-get install mupen64plus
Then, the Mupen64Plus executable should appear at /usr/games/mupen64plus. Mupen64Plus-Qt should auto-detect this the next time you run it.