Advice for lower power clock type project
I will make my comment an answer:
Try one of Arduino supported RTC's boards. The most popular (and simple) are: DS1302, DS1307, PCF8563, and - very nice, precise DS3231 (my fav). Hook any of them up to your Arduino, add some (readily available) libraries, tweak a little and voila. The RTC will work from coin battery (or high capacity capacitor called supercapacitor, like this one)  

Most (if not all) of mentioned RTC have 'alarm' function: you can set up your system to wake up your Arduino from sleep (sleep state to save battery) and gradually light up the LED. 

To limit current of Arduino try removing LED, or - better - try to use Arduino Pro Mini. To modify it for better power saving see here  

EDIT:
There are some other low-power boards:
Moteino
Rocket Scream Mini Ultra
JeeLink  

Also, (as mentioned in one of comments) you can try one of ESP8266 and push it to one of many sleep modes. Try this link.