Pins 1 and 8 are shown on the left diagram merely to orient you. They are just two of the segment connections. This is a top view. Pin 1 is underneath, below the + sign. The pins are numbered clockwise in this top view.
Connect pin 3 or 14 to the Arduino +5v supply pin.
Use a resistor to connect pin 1 to GND and segment A will light up. (One end of the resistor to pin1 and the other end to GND).
Use another resistor to connect pin 13 to GND and segment B will light up.
Use a third resistor to connect pin 10 to GND and segment C will light up. etc.
Don't try to simplify things with a single resistor in the Anode lead, you need separate resistors in the Cathode leads to get good results when multiple segments are lit simultaneously.
After you get it working this way you can use your Arduino I/O pins instead of the GND connections to control your display via software.
EDIT: To answer your next question ... 220 ohms or more would be a good choice.
ALSO: NP means there is 'no pin' at the expected location, NC means there is a pin but there is 'no connection' between the pin and anything inside the device.