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.