Running stepper motor at a set time interval

How accurate do those times of day need to be? The Arduino by itself is a poor time keeper in terms of time of day. Use a Real Time Clock (RTC) to keep time. Then it is not hard to compare the current time to the required turn on and turn off times and take appropriate action. There are libraries for each of the RTC modules that will help with that. I like the DS3231 RTC modules.