When you don't draw a lot of current, you can run the Arduino from the Lipo cell while charging.
But there are many things you have to know to do it safely.
Sometimes a Lipo battery pack has a protection circuit board inside. That is prevent shortcut, overload and too low voltage.
With such batteries, the charger doesn't have to stop charging when the battery is full.
A good charger will stop at 4.2Volt, because that is the maximum that a Lipo cell can have.
If you keep the Arduino running with an empty battery, the Lipo cell gets damaged below 3.0V.
That might cause internal shortcuts and that might cause dangerous situations.