Using a phone battery as a power source possible?
Sounds like a good solution. 

Just note that most cellphones use Lithium batteries. Those are typically around 3.7 volt. They require some special way of charging, or they may explode or catch on fire. They also get damaged if discharged too much.

These batteries may have some protection circuitry inside them. This would be ideal in your case, as it protects against over-discharge, and accidental shorts.

An alternative would be the cells used in notebooks called 18650s. These are available with protection and without. The advantage is that you have a standard size. You can buy battery holders for them. That way you can easily exchange batteries. You can also combine cells to create higher voltages.

As for the runtime; you can make a rough calculation of it.