Ok so the min hour thing is wrong on the crontab. It should be something
like this:
00 06 * * * [ `date -d tomorrow +%d` -eq '01' ] && /the/script
for 6:00 AM on the last day of each month
I didn't know it was possible to put this kind of thing (the output of a command) in crontab. I searched for it in the man page and could not find it ( man 5 crontab on a Gentoo system). Interesting indeed.