Q:

Charge 3.7 Li-ion Battery with 5V battery charger? 

Is it possible/save to charge a 3.7V Li-ion Battery with a 5V battery charger like this one: https://www.amazon.de/gp/product/B0191EVW0C/ref=oh_aui_detailpage_o00_s00?ie=UTF8&psc=1
Additional question: Would this be sufficient for an Arduino Pro Mini 5V using the VCC Pin? 
Thanks! 

A:

Yes, this charger is okay for a single 3,6-4,2V li-ion cell.  An ATmega328 will run fine at this voltage when you obey the clock speed limit (section 29.3 of the datasheet).  Make sure to measure battery voltage using the Arduino, so you don't discharge the battery too deep.