There are lots of circuits out there to have an Arduino switch off itself. Usually built with two or three transistors, look for latching power supplies. You use a push button to switch it on, the same button or a signal from the Arduino to switch it off again. These circuits are quite simple and cheap to build.
To read the voltage of the battery you need a voltage divider to bring down the maximum voltage your batteries can produce (normally 4.2-4.3V for a nominal 3.7V battery) to <1V and use the internal 1V reference for the ADC to read the actual voltage. Note that you can't use the default reference, as that'd be the very voltage you try to measure. Again a very cheap circuit, as all you need is two resistors (use large resistors as otherwise you're leaking too much voltage - 500k total or so) and a small cap for stability (between the pin and GND - 10-100 nF should work for this).