Yes, no… that kind of battery has nothing but the bare cells in series. It is incumbent upon the user to cease drawing current when the voltage gets too low.
I stop when under load (heavy load) the voltage goes below about 3.3 volts per cell, and I look to have a recovery voltage of 3.7 volts.
This leaves a lot of power unused, but yields very good battery life.
Similarly, I charge to 4.17 volts, every 0.1 below the nominal 4.2 permissible again is good for the cell, at the expense of operating time - so I live with that or use a larger battery than I would if I was willing to keep buying new ones, and take risks at the low end that I am not interested in. Taking.
Now I drew you a latching relay circuit but can't get it into this post at this time, but it is simple, just a pushbutton in parallel with the relay contacts that supply all power. As soon as you push the button, the Arduino take over by energizing the relay, and continues to do until it's time to shut down, when it just releases the relay and everything is dead.
An analog input and as you say appropriate voltage divider is a perfect way to keep an eye on the battery. Obsessive types might even go so far as to monitor every cell; with good batteries well cared for and always balance charged, it should not be necessary… my batteries end up balanced after use, when they don't I know it is the beginnings of the end for a cell and therefor the battery. I do not do repairs on lipos.
See battery university dot com or something like that, it has more than you could ever want to know about all kindsa batts.
One last must know fact: if you aren't going to use a lipo, discharge or charge it as you must to achieve 3.7 volts per cell, some say 3.8, but the point is neither charged nor depleted. Leaving a lipo charged for a day is about like putting a full cycle on it I have seen said, dunno for sure but it is a good idea even if a PITA to charge, then use, then balance charge to the storage voltage.