I would suggest an Arduino pro mini from ebay (around $2) you still need a wall wart to power it and (more importantly) a serial interface cable to program it if you don't have one or can't cobble one together but I expect that will be your cheapest option. Its hard to beat a board with reset circuit, crystal osc and programmer assembled for a couple of bucks. For me (because I have a PIC programmer) a 12f series PIC in 8 pin dip would be an option (internal oscillator) but the chip from digikey is almost as much (more with $8 shipping) than the nano from Ebay and the programmer is expensive. While you may be able to use the Ds3231, you still need a micro (at least at some point, and possibly at each power up, I haven't ever used one in this type of application) to set the registers to start the alarm function working. I suspect the alarm function may not survive a power outage, time keeping does but I suspect all the other stuff shuts down to save power (but I'm not sure, you would have to experiment with one to see). If you can survive a power failure (or supply the mini from a cell phone charger type device using an 18650 lipo battery as a UPS again a couple of bucks on ebay) you could do without the rtc. No reason you couldn't write code on the arduino to make your own rtc as long as you can keep it powered for long enough during a power failure for your application.
Peter