Stepper motors do not have to be held on
The whole system will be running from a battery, so by putting the arduino to sleep I am hoping it will stop sending power to the stepper motor (the stepper motor is powered directly from the arduino).
Stepper motors don't have to be held on. You can step them and turn off the current. Of course, this means that if something external moves the stepper, the arduino doesn't "know" about it - but that happens anyway if you put the arduino to sleep. Your display board - whatever the moving part on it may be - will need some sort of reference marker that can be picked up by a sensor … although I suppose you could zero it and just rely on the stepping, or have some kind of "lock" solenoid. If the stepper connects to a gear train (eg, worm and gear), the friction in that will probably be enough to keep things in the right spot.
But seeing as you want to use a real-time-clock anyway (correct choice, BTW), that clock can generate a "wake up" pulse every so often. The arduino wakes up, checks the time, goes straight back to sleep.
Generally, you don't run any kind of motor "directly off the arduino". First, they draw too much current. Second, the inductance of the coils generates back EMF when the coil is switched on and off, which tends to fry logic electronics. Easiest solution is to buy a steeper motor shield, or to use a driver board. You can make your own - I've done it, it's not that difficult for a small motor. I used a ULN2003, which will happily switch 24v - 12 was enough for my application.