One of the reasons to choose the Arduino Pro Mini is because I've used it before in battery power projects and I know I can reduce its consumption a lot so you I don't have to recharge my smart devices every day. Let's calculate:

MPU6050 datasheet https://www.invensense.com/wp-content/uploads/2015/02/MPU-6000-Datasheet1.pdf  4mA (3.9mA) or less

Arduino Pro Mini power consumption test https://www.iot-experiments.com/arduino-pro-mini-power-consumption/ from 13mA to 3.5mA or less

So assuming 1800mAh (a regular rechargeable LiPo battery 1x2 inches)

1800/ (4 + 3.5) = 240 hours

240 hours/24 hours in a day = 10 days.

So 10 days it should last the bottle, working 24h on a single charge, without too much optimization. 

But we can make it even better, by powering down the Arduino (eg: once the LED is ON). 
Or removing the status LED from the Arduino or the Accelerometer. Or disabling some features from the MP6050 (accelerometer , gyroscope , DMP)