Power analysis of a simple circuit and battery lifetimes
If you have a multimeter you can calculate dc power based on voltage measurment and current measurement but i think you are more interested in battery life? If so, connect the battery and measure the current taken. Then compare this to the ampere-hour figure of your battery - if you have a 1Ah battery this can provide 1A for 1 hour or 100mA for ten hours or 10mA for 100hours. But battery suppliers usually assume the Ah rating is for all the usable energy in the battery. It's probably more likely you'll be able to use about half of this but... and here's the small print....

How close is the battery terminal voltage to what your arduino needs? If you are directly powering it, the output voltage may not have to droop much before the arduino gives up. You maybe be feeding it via a linear voltage regulator and this will give you a bit more headroom. If you are using a switching PSU this will give you more. I can't say - only you can.