I would for sure not jump into panic together with the customer as it would not help us. Just by saying: “Ok I am on it, can bring the customer into a more calm state”. If there is no recent back up, I would make a backup first (files + database) before we do any intervention (to make sure we have a way to go back).
I would check the html source to see if I see a clue over there.
I would try to detect on which domain the error occurs. Is it a database problem, a webserver problem, a capacity problem, … .
Is the backend available (through /wp-admin)? If it is and there is a caching plugin, I would purge the cache.
If there is a non-Wordpress problem, that should be tackled first. Sometimes a simple restart of the server could do the trick (when possible of course).
When the ‘outside’ does not leave any clues. I would check if we can login into the database directly (typically through PHPmyadmin). If there is no db username / password, we have to find this in the wp-config.php.
If that seems to be working, I would through e.g. FTP (or online file explorer) create a simple PHP file (just echoing something) : e.g. this_is_a_test.php and see if that file is available through a browser.
If everything seems to be working, I would off course consult my friend Google as well (with the clues I gathered so far). If the time spent on the analysis phase is taking quite some time without progress and/or a solution seems to be far away. I would consult a co-worker (if the policy foresees that).
If that does not bring us any further, I would check when there was a recent backup made. Next, I would check with the customer if it’s ok to restore the backup of that time. How that happens depends on the backup system in place (I often use Updraftplus or straight re-uploading set of files and re-populating the corresponding database)