If you can't do scheduled tasks, then one possibility (a workaround, really) is to use a log table that tracks when the last update was done. Whenever any page is loaded from the site, check the log table, and see whether or not -- based on the date -- the database needs to be updated. If it does, then you can process the update.
You can also use application variables in conjunction with the log table to prevent needing to go back to the database repeatedly -- once one session in the current application cycle confirms that the database is updated as it needs to be, the application variable can be set, and the system doesn't need to go back anymore while that variable is set (I like to set the date/time that the application next needs to be updated as the variable -- that way, it knows both whether it is set, and if it is set, whether the next refresh is due. Much better than simple boolean.)
It's still a good idea to use log tables, though, as they prevent the system doing everything all over again if the application restarts for any reason.
However - how viable this is as an option is really heavily dependent on the specifics of the solution you're trying to implement, though. If the specific system state at 12:01am is a factor, then this may not work.
If you're using a Windows PC, though, another workaround -- provided that your PC is always on at 12:01, anyway -- would be to set a scheduled task on your local machine that would load the page, whether or not you were there. In Control Panel --> Administrative Tools, you should find something like "task scheduler" or "scheduled tasks". You can set a task to run at 12:01 daily, with a command along the lines of this:
C:\Program Files\Mozilla Firefox\firefox.exe http://www.sitename.com/updatepage.php
This would cause Firefox (or another browser, depending) to fire that page up and trigger the refresh. Again, it's not ideal, as it means your PC needs to be on until then as a minimum, but it's an option if you can't use proper scheduling on the host. I'm not familiar with other OSs, really, so I can't swear to whether or not something similar can be done.