Assuming you want your system to work as long as possible with the battery, you should consider changing your elements :
- doDo NOT use Arduino Uno or similar, but go for a Pro Mini (3V3/8MHz) with no LED.
- selectSelect a low consumption display. e-ink are best, as they do not consume anything as long as you don't change the display.
- lookLook at "sleeping" capabilities of your system. There are a few libraries or articles explaining how you can put your system to sleep and only wake up every x minutes.
- considerConsider using a 1.5V battery and a boost DC-DC converted.
One AA battery will last weeks if you are following the above recommendations. I have one such system running since 10 years at home, and I change my battery roughly every year.