The EVE Online launcher should update automatically every time you open it.
To start with, please ensure the launcher is fully closed, instead of only minimized, and launch it again.
If you are not sure if the application has been fully closed, then try restarting your computer and running the launcher again to see if it updates by itself.
If your launcher is not starting the self-updater, you can try starting the process manually by pressing the green Update Ready! icon in the top right corner of the launcher.
If that still doesn't update your launcher, you will have to reinstall the launcher. It is possible to reinstall the launcher without downloading the entirety of the shared cache files again. See the following page on how to do this:
If your launcher’s update progress bar does not start, the launcher may be stuck installing the downloaded files, or it’s constantly updating files. It may also be having issues downloading or installing the downloaded files.
Please try the solution below to see if that resolves your issue:
Verify security and privacy software blocking the launcher update
It is possible a security or privacy software (antivirus, proxy, or VPN) is blocking the launcher update. You can check any third party software that may be preventing the update of the game files by the launcher by adding security restrictions or modifying your network connection to our update server and add it to the exceptions list or disabling this software while updating the game.