sudo systemctl status shows you what services are running, sudo systemctl list-units lists everything, including the services that failed. sudo systemctl status gdm.service shows you the status of one service in particular, and sudo journalctl -u gdm.service shows the output/logs of that service.

There’s a decent chance something is not starting due to misconfiguration. I’m guessing GDM based on previous comments. You can also check /var/log/pacman.log (make sure to save a copy, just in case), to see what packages changed/updated.

If you think it’s a pacman issue somehow, you can reinstall your entire system (excluding AUR or self-made PKGBUILDs). Note that this is almost never required to fix an issue. In a properly working system this “shouldn’t harm anything”, but nothing can be guaranteed on a broken system. The command for that is sudo pacman -S $(pacman -Qqn)