A typical e-paper display consumes no power after update ( 0, zilch, nought :) ).
For update again a typical display 2.5mA for 2.5seconds to completely refresh the screen and if refreshed every hour, gives a daily power consumption of 0.042mAhr.
The ESP8266 requires 75mA for 10-secs to get updated data and at an update every hour, requires 5mAhr
Therefore a 2600mA battery (e.g. 3x1.2v Nicads in series) would last for 22 days.
To change all this to 5-min updates would result in a daily demand of:
Display : 0.52mAHr
To run it all for 1 month at a 5-min update rate would require a battery of 60.042 * 24* 30days = 43Ahr battery, or something like a car battery.