The line in red is the problem.
> Warning: fopen(/var/www/web588/html/schule/moodledata/cache/languages): failed to > open stream: Permission denied in /var/www/web588/html/schule/moodle/lib/moodlelib.php on line 4543 Running backups if required...
It looks like the user the cron job runs under cannot open the necessary files. You need to set the job to run as the same user as the web server runs as (probably "www-data" or "nobody"). That way it'll have the same permissions as if you started it from the browser.
Gavin
Installing and upgrading help
Installing a Crontab
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