The Arduino will need to do both, but this is no problem. It needs to run the H-bridge and also communicate with the RTC.
The RTC will provide time of day and date information, but I'm not clear whether this is actually needed in this case, or if the Arduino just needs to use it as an accurate time base to drive the clock coil. Most inexpensive quartz clocks I think just pulse the coil once per second, but I'm not sure in this case. It that what is needed, or something else, a faster signal?
There are a couple options. To use the RTC just as a time base/frequency reference, it has an output pin that can provide a 1Hz (one pulse per second) signal. (There are also several other frequencies that can be selected.)
OTOH, to use the RTC to track time and date, I like the Time library, http://playground.arduino.cc/Code/Time
The Time library implements a software RTC and then synchronizes it with the hardware RTC (Chronodot) on some given schedule (every five minutes by default).
PS: The Uno does not really have a "clock module" in the sense of an RTC. It does have a ceramic resonator to provide the system clock. All microcontrollers need a system clock of some sort, and there are many ways to provide it. Note that the system clock is not a time-of-day (real-time) clock, it is simply a regular signal that the MCU uses to synchronize all its internal logic. As noted earlier, the ceramic resonator is not all that accurate for timekeeping purposes.
There are a lot of options and a lot of flexibility, so take some time to understand what is possible and figure out what the best approach for driving the clock is. For instance, the Time library can function perfectly well without an external hardware RTC, using the system clock for its timing. If the system clock were derived from a more accurate source (e.g. a crystal), this could provide decent timekeeping, but even most commonly used crystals will still be an order of magnitude less accurate than the Chronodot/DS3231. Have a look at the DS3231 datasheet
so you know what you're dealing with and what its capabilities are.