Hi there! After installing the game and all required libraries (32bit versions, of course), I was able to run it. It blinks with a black screen for a moment, than I see this in my terminal:

Absolute executable path: /home/vj/GOG Games/Grim Fandango Remastered/game/bin/GrimFandang
Leaving working directory as: /home/vj/GOG Games/Grim Fandango Remastered/game/bin
Not found: 'x86/shaders/compiled/deferred_light_v.glsl'
Not found: './Saves/registry.sav'
Font 'TREB13BS.LAF' loaded; 'FontsHD/TREB13BS.LAF.txt' -> 'FontsHD/FuturaStd-Heavy.ttf', 38.000000px
Font 'COMICSANS18.LAF' loaded; 'FontsHD/COMICSANS18.LAF.txt' -> 'FontsHD/FuturaStd-Heavy.ttf', 52.650002px
Font 'KINO14S.LAF' loaded; 'FontsHD/KINO14S.LAF.txt' -> 'FontsHD/FuturaStd-Heavy.ttf', 40.950001px
Font 'OCR10.LAF' loaded; 'FontsHD/OCR10.LAF.txt' -> 'FontsHD/OCRB.ttf', 22.000000px
Font 'KOLOSS52.LAF' loaded; 'FontsHD/KOLOSS52.LAF.txt' -> 'FontsHD/Koloss.ttf', 52.000000px
Font 'KOLOSS69.LAF' loaded; 'FontsHD/KOLOSS69.LAF.txt' -> 'FontsHD/Koloss.ttf', 69.000000px
Font 'KOLOSS100.LAF' loaded; 'FontsHD/KOLOSS100.LAF.txt' -> 'FontsHD/Koloss.ttf', 100.000000px
Font 'MEREDITH46.LAF' loaded; 'FontsHD/MEREDITH46.LAF.txt' -> 'FontsHD/Meredith.ttf', 46.000000px
Font 'MEREDITH38.LAF' loaded; 'FontsHD/MEREDITH38.LAF.txt' -> 'FontsHD/Meredith.ttf', 38.000000px
Initializing Theora Playback Library (1.0 RC4)
- libtheora version: Xiph.Org libtheora 1.2.0alpha 20100924 (Ptalarbvorm)
- libvorbis version: Xiph.Org libVorbis 1.3.2
------------------------------------
Creating video from data source: MoviesHD/logos.ogv

After that it immediately consumes all the CPU at 100% and fans start spinning like crazy. It looks like it renders or compiles something really heavy.
I waited for about an hour and then killed the GrimFandango process.

Tried several times. Sometimes black screen stays, and CPU is not spinning, but most often I observe behavior showed above.

I'm using Arch Linux. Did anybody face the same issue? If no, maybe you have some suggestions for me? Anyways, big thanks in advance!

UPD: maybe I should also note, that I only have an integrated Intel GPU, and my CPU is i7 8th gen

Well, just a couple minutes after initial post I found a grim-fandango package in AUR which told me that I have to install couple more libraries:
lib32-libxcursor
lib32-libxinerama
lib32-libxrandr

Seems that they're not in game dependencies in the store page, nor they are listed as a dependencies for actual game dependencies. After installing those game runs smoothly. Hope it'll be helpful for anybody

Apparently, I can't mark my own reply as a solution, so if anybody can post anything here, I'll gladly mark it as "SOLVED" :)