How to connect a Sparkfun LiPoly Charger to an Arduino Nano?
Just connect the system out to the Arduino Pro (BATT connector), as shown in your picture.

The Arduino Pro uses 3.3V for the ATmega328 MCU, but the board (the Arduino itself) has an on-board regulator which can accept up to 12V DC.

Check out the specs on the item at Sparkfun: