Also, you should be more clear how you plan to multiplex these.
1 segment at a time? You can get by with Arduino driving the anodes, and shift register sinking the cathodes. Need 1 current limit resistor per anode.
1 digit at a time? You will need 4 PNP transistors, 4 base current limit resistors, and 7/8 cathode current limit resistors.
Little more hardware, but the multiplexing will be faster & brighter.