The file command needs to be run on your actual epsxe binary, you simply ran the command as I posted it. I do not know where your epsxe binary is on your system, you will need to either figure that out on your own and then run the file command on it, or link to whatever guide you used specifically to install epsxe and I might be able to figure it out.
If you absolutely cannot find it, one of these should help:
As for the results of your system, you are running the 64-bit version of ubuntu. If your epsxe binary ends up being 64-bit, installing the library I originally linked to should work. If it's 32-bit, we will have to install the 32-bit version using dpkg --force-architecture.
find / -type f -iname epsxe
EDIT: If you used this guide, it looks like it is in /usr/local/games/epsxe/ directory. There's a bash script that launches it at /usr/local/bin/epsxe. It also looks like that shell script sets the LD_LIBRARY_PATH to /usr/local/games/epsxe/, so you could probably place the libgtk-1.2.so.0 in /usr/local/games/epsxe/ without any fuss.