You probably downloaded the .exe version of Neoalpha. You need to download the .jar version. It is also possible that you installed a Java version older than 1.7.0 (7).
After launching Neoalpha at least once the files should be located at /home/yourlinuxdistusernamehere/.neoalpha. If you don't see the folder, you must show the hidden files (that option is located in the up-right corner menu) (note: this could vary between Linux distributions). You can also use the 'Open texture pack folder' option in 'Mods and Texture Pack'.
In the terminal go to the directory where the Neoalpha jar is located and then type in 'java -jar Neoalpha.jar'. Neoalpha should launch.
In order to play Neoalpha on Linux you'll need to install the Zulu build of OpenJDK. OpenJDK itself won't launch Neoalpha. You can find these builds at this link. Note that the '.deb' files can be installed only on Debian-based distributions (e.g. Ubuntu, Linux Mint, Pop!_OS) meanwhile '.rpm' files can be installed only on Fedora Linux-based distributions (e.g. Red Hat Linux, Oracle Linux, CentOS). Note that Neoalpha may not run if you have multiple Java versions installed!
You probably have multiple Java versions. Uninstall other Java versions (leave just one). From what I've heard (not sure) newer Java versions are unsupported on older Minecraft versions. Java 8 is guaranteed to work. If none of these are the case, you can try updating / installing graphics card drivers. Replacing the LWJGL files also supposedly fixes the issue (instructions on how to do that below). I can't guarantee that these solutions will work though.
Go to the LWJGL 2 website (link here) and click 'Downloads' on the left side. Click the 'LWGJL 2.9.3' link. You will be redirected to SourceForge. Download 'lwjgl-2.9.3.zip'. Open the '.neoalpha' folder (should be located at /home/yourlinuxdistusernamehere/.neoalpha/. If you don't see it, you must click the 'Show Hidden Files' option.). In the '.bin' folder replace 'jinput.jar', 'lwjgl.jar' and 'lwjgl_util.jar' from the zip file (these files are located in the zip file at '/lwjgl2.9.3/jar'). In the 'natives' folder ('/home/yourlinuxdistusernamehere/.neoalpha/bin/natives/') replace all of the files with the files from the zip file (files from the zip are located at '/lwjgl2.9.3/native/linux').
You don't have graphics card drivers installed or your drivers are outdated. You need to install / update them. I'm not going to write a guide on how to do that, as the installation depends on the graphics card and the Linux distribution. Just search up 'how to install (your graphics card developer/name) drivers on (your Linux distribution name)' in a search engine.
Either something is wrong with your internet connection or the server is offline. To check if the server is offline just go to the Neoalpha website. The server status is at the bottom of the Home page. If it is offline it should say 'The main server is currently offline. This doesn't stop you from joining other servers though!'
Your internet connection could be slow. Try e.g. resetting your internet router.
That probably happens when you created an account very recently. You should wait a while to play online.