Can you charge and use a LiPo battery at the same time?
can I charge the battery while the arduino is still on, but on power savig mode?


You can draw power from a Li-Ion battery whilst charging it, but it isn't recommended. This is because the load on the battery affects the ability to detect the different phases of charging, and determine the end-of-charge condition.

Instead you should provide a bypass facility so that if external power is applied the external power is routed to the Arduino to power it, but when the power is removed the battery provides the power to the Arduino.

The circuit I usually use is:







simulate this circuit &ndash; Schematic created using CircuitLab

When 5V is applied the battery charges up. Q1 is held off. The 3.3V regulator is then powered through D1. When the 5V is removed Q1 is turned on by R1 pulling the gate low and the battery powers the 3.3V regulator.