Again, as said by the OP, my solution already worked. Even Mojang recommends using Oracle's Java mentioned in the download page for best results.
Unfortunately, this is not possible to do so. There's only an executable version for Linux and others, and that's it. No installer or anything; you just download it and run it. The easiest way is to just put the .jar into a desktop, open the terminal and input these commands:
cd $HOME/Desktop chmod +x Minecraft.jar
The last command will make the .jar executable, so you will be able to launch it with just a double click on the file. That's the real easiest solution, so you're on your own if you want it to still put it in the Dash screen or similar something (assuming you're running the normal Ubuntu).