With the current GOG version v1_1_4_26003 one will only get a 32 bit version.
So I asked the developer and was informed that it should have the 64 bit version and got the hint about a script providing it otherwise:
* https://itch.io/post/629039
which still works.
And this nice little game really deserves to get provided in best shape ... incl. 64 bit binary.

I will give the commands with explanation for those wanting to know how it works - but it does what is done by the script plus changing the GOG `start.sh' file to call the 64 bit version only.

# * Now step by step howto (tested with GOG version v1_1_4_26003):
cd <Path_and_GOG_Little_Bug_Install_Dir__User_Selected>/game
# ... changing to game directory - incl. `Little\ Bug.x86' & `Little\ Bug_Data'.
curl -O http://download.unity3d.com/download_unity/472613c02cf7/TargetSupportInstaller/UnitySetup-Linux-Support-for-Editor-2017.1.0f3.exe
# ... fetching the Unity file archive (of appropriate version).
7z x UnitySetup-Linux-Support-for-Editor-2017.1.0f3.exe
# ... extracting it - creating directories '$PLUGINSDIR' and '$INSTDIR$_59_'.
cp \$INSTDIR\$_59_/Variations/linux64_withgfx_nondevelopment_mono/LinuxPlayer Little\ Bug.x86_64
# ... copying binary with game name to game directory.
chmod a+x Little\ Bug.x86_64
# ... making file executable for all (i.e. user, group and others).
/usr/bin/cp -Rdpv ./\$INSTDIR\$_59_/Variations/linux64_withgfx_nondevelopment_mono/Data/* Little\ Bug_Data/
# .. copying all useful Unity files incl. mono library for x86_64 to the expected location.
rm -rf \$INSTDIR\$_59_ \$PLUGINSDIR
# ... deleting recursively the no longer used directories from extraction.
rm UnitySetup-Linux-Support-for-Editor-2017.1.0f3.exe
# ... deleting the downloaded Unity archive file.
cd ..
# ... going to GOG game directory chosen by running the GOG installer script
# `little_bug_v1_1_4_26003.sh'.
vi start.sh # or another editor of choice replacing vi
# changing line:
./"Little Bug.x86"
# into two lines (first is old and commented, second is new and active):
# ./"Little Bug.x86" # old 32 bit binary
./"Little Bug.x86_64" # new 64 bit binary

I hope this helps - as the 32 bit binary should no longer be used on current PCs.

I would like GOG to make such heavy liftings for its customers, but GOG seems
not interested to create install scripts for Linux users which would be best for gamers.
Many `Windows Games' or `32 bit Linux ones' do have 64 bit OSE with current HW support
running much better than the original and as it is legal GOG should be able to provide it,
e.g. for Aquaria.
Formerly, GOG even created manuals as PDF for games they offer ...

Additionally I would wish for QA that the given platforms are really supported
on the same ground - all DLCs available for all platforms.
And to make sure that 32 bit SW is specially marked as being totally outdated and a risk
of no longer running in near future - macOS was first, but Linux may follow very soon ...
So GOG should have solved this issue before any user can even spot it.
That care should be given on the product pages:
current version, (really) supported platforms, (working and configurable) controller support,
correct HW requirements of current version ...
And frequently this is just wrong or outdated - even the genre may be wrong.
So reading reviews and forum contributions is more informative for me than
the current state of the product pages in general.

There are some gamers wanting to help in improving the experience ...
may be nice to get heard and seeing that things do improve.
My lessons learnt are that one can currently help best by forum contributions and reviews ...
But at least the stream of updates is flowing after some stagnation ...
and that in single player mode GOG games are DRM-free.
Would wish for more ... but that's the way it is - still better than Steam
(as long as GOG stops providing critical components like downloading development versions -
for helping game development - only via galaxy client [being data collector and DRM-like]
and not answering support tickets).
There is way for improvements ... just waiting for it. ;)