There are already a lot of guides on how to set the resolution of the game to Full HD. It includes downloading the Spanish launcher and/or setting commands in launch properties. That method didn't work for me, and it also messed up my menu as I lost half of the options like settings, load, etc. So here's the different solution, which I hope will help those who couldn't get Full HD to work in their game. 

The game runs in 640x480 resolution by default. Launch.exe lets you change the resolution of the game. But, the launcher that comes with the Steam version wasn't launching for me ("Error: couldn't find alone4.exe") 

The following is a corrected version of launch.exe for Steam. It also comes bundled with a partial fix for the game not running on Windows 8 or 10, as well as higher-quality movies ripped from the Dreamcast version.

1. Download Alone In The Dark TNN Fixes & Cutscenes [app.box.com]
2. Delete the movies folder inside your game folder.
3. Extract everything from the archive into the game folder and replace as necessary.

Now, the launcher will work and you will be able to set the resolution to 1080p, but you have to launch the game from the launcher (launch.exe), not steam.

You did it. Now the game will run in 1080p with HQ cutscenes!