You can try. First thing to check is if the OpenGL driver is in the Ubuntu MATE ARM repos (I don't know if it uses the Raspbian repos or only its own ones, in which case they might not be up to date). That's the only part that could be different between Raspbian and Ubuntu MATE.
To be sure, you can try the following on your Pi:
apt-cache policy xcompmgr libgl1-mesa-dri libalut0 libalut-dev mesa-utils
For each one, you'll see if there is a "candidate" version. If that's the case, it means that the packages are in the repos. If not, either they're missing or they have a different name (but I doubt that Ubuntu would repack packages related to Mesa with a different name).
The author of the article is providing ARM versions of the missing libs. Note that they're on a Dropbox account and the article is getting a bit old, so check if he hasn't removed them in the meantime. If they're not available anymore, I assume that they're free/opensource and could be compiled from source but it might not be the easiest thing to do, especially if you're not familiar with this.
I don't know of a Minecraft PPA with ARM builds but it doesn't really matter: the only point of the PPA is to install Minecraft easily but it doesn't do anything special apart from checking that Java is installed, downloading the .jar and providing a launcher. If you manage to install the game manually, it's easy to make your own launcher afterwards.