Best way to deal with voltage loss in batteries
Each battery have own discharge behaviour:



The part of graph where it's more acclivous - it's the life cycle of battery. So real voltage of battery will drop down in short time after battery plugged into device and for good battery (ex.: E2 on the picture) it will save this voltage on the most part of life cycle.

So if you need to power device with voltage in range 3.7-4.2V - you need to select battery by higher threshold as minimum. Battery capacity about 100mAh is too small, even if you will use strongly power save (with consumption about 5uA) beetween two measurements. I use battery 3.3V 1000mAh for power board with STM8L151 series MCU and with all my efforts to power saving it will work only about 1 year.

If it's difficult or unreal for arduino to achieve good power save characteristics, so you need battery with big capacity and may be, it will be easy to use some linear regulator as LM317 series and 9V battery.