Make sure you have wine installed at minimum 2.21 version (it was tested by me). Here is how to achieve this: latest wine installation
Make sure you have proprietary producer's graphic card drivers installed. For example for NVIDIA required version is minimum 375.26 (it was tested by me). Here is how to achieve this: NVIDIA GTX GeForce drivers installation
Below there is a step by step instruction how to install and run the game:
env WINEPREFIX=~/.wine32-oddworld_1_abes_oddysee WINEARCH=win32 /opt/wine-staging/bin/winecfg
In a popup window in Applications tab select Windows 7 option. Next switch to Graphics tab, check Virtual Desktop Emulation checkbox and fill in your desktop's resolution in use, for example 1366x768.
cd [directory that contains the installation file setup_abes_oddysee_22.214.171.124.exe]
env WINEPREFIX=~/.wine32-oddworld_1_abes_oddysee WINEARCH=win32 /opt/wine-staging/bin/wine setup_abes_oddysee_126.96.36.199.exe
It will launch installation process that installs the game. After successful installation it will create shortcut on your Desktop that allows to run the game.