If you close Outlook, what program will send the auto reply message? You would need some other email program to check your email and send automatic replies. If you leave at the same time every day you could schedule another email program to start at a specified time each day through the Windows Task Scheduler. You could then stop the program manually when you arrive the next day or you could schedule a script to kill the process for the other mail program at a specified time each morning.
Or you could create a batch file that runs a tasklist /fi "imagename eq outlook.exe" command periodically that will start the other email program if it doesn't see Outlook.exe running and a taskkill /f /fi "imagename eq otherprogram.exe command to kill the other mail program when it finds Outlook.exe in the tasklist. But you aren't going to be able to set an Outlook rule to send automatic replies when Outlook isn't running to process the rule.
If you are using a Microsoft Exchange server with Outlook, you can specify automatic rules to be run by the Exchange server, e.g., if you have a webmail interface to the exchange server through Outlook Web App, then you can configure an auto-reply to be sent by the Exchange server. However, that only gives you the capability to configure automatic replies from a start date and time to an end date and time. It doesn't provide the capability to specify that the automatic replies start at a specific time each day and stop at a specific time the next day.