After doing initial enumeration on this room I found the /cloud directory. Here it was possible to upload image files. To bypass the file extension filter, I simply added a null-byte to the path.
That is, if my payload was called shell.php, I type the URL like this LOCAL_IP:PORT/shell.php#.jpg
This way it was possible to gain initial access.
KeePass cracking
Again, simply doing basic enumeration showed another user "sysadmin".
I searched for files belonging to this user with find / -type f -user sysadmin 2>/dev/null and found a KeePass file called dataset.kdbx.
This file was extracted by setting up a python3 HTTP server at the target - python3 -m http.server 4444
and once keepassxc was installed, I could get the password of sysadmin.
Root
I noticed that the /home/sysadmin/scripts directory was owned by the root user and could kinda guess that the script.php would be executed by the root user a some point.