No more. Feel free to say. Felt a bit lonely while solving and after solved the problem. Do not still know why I couldn't write to /var/cache/apt and why it became read only. I found out that it could be a one corrupted file with a wrong character somewhere, or caused by interrupted installation of an app.

Tried the mount -o remount,rw / (which was new to me) and it could have been solved with some chown or chmod stuff (possibly). Didn't come into my mind to try resetting with boot.

Remounting read/write should work if the filesystem had been changed to read-only. You should first check if that was the case. (can be done by mount command and checking the flags)

Similarily; first analyse what the situation is and then try the correct remedy. Of course in many cases reboot will fix it but that's a kind of blunt weapon IMHO...