Yes, you can run a cron script... but if the main site goes down, do you really think that another folder on the same server is going to stay up? I don't know about your particular site, but for most sites I would assume that failures are likely to be due to events impacting the server computer, not a particular folder on the server, such as overloading, power outage, disk or other system failure, etc.
As for the implementation of the cron script, you should take a look at rsync, which can synchronize folders both on the same computer or between computers. Another possibility is to store your website in a Subversion repository or other SCM, and simply run an update command periodically. Of course, this requires you to have some form of hosting for the Subversion (or other SCM) repository.
In cron, you can use the abbreviation "@midnight" to indicate that an event should occur at midnight (instead of having to fill out each of the various time fields). See the cron Wikipedia page.