Electronic – arduino – Working with a cathode-based LED seven-segment display

You are not using enough voltage.
The page you referenced notes that there multiple LEDs in series.
They say:

• Large (58mm) 7-segment LED display
  Digits color can be either red or green.
  Type: 12101BEG
  If max: 30mA
  If peak 1/10 dutycycle, 0.1ms: 100mA
  Vf red: 7.2-8.8V
  Vf green: 8.8-10.0V
  Each segment consists of four LEDs connected in series.

In most cases simply trying every combination in turn will work.
There is a small chance that you will destroy segment LEDs with reverse voltage.

Supply two leads from psu via a resistor so you can easily swap both leads.
Try each combination in turn.
A1 A2 A3 A4 A5 B1 B2 .......

The red wire is hooked up to the 5v output on the arduino and the blue to the ground. As can be seen the decimal point has been lit up in green, and if I move the red wire four pins to the left the decimal point turns red instead.

As they say CA = common Anode = common +ve you may have
2 x Anode's
+ 8 x Cathodes.
So Bottom right may be common green anode and
bottom left = common red anode.

SO increase Vsupply to say 8V, use a 10k resistor for initial safety, put V+ on bottom right and try all pins with v-. If that works then put V+ via 10l on bottom left and repeat.

The supplier should be able to provide a datasheet.

The purpose of the resistor is to limit current to a segment LED, by dropping the surplus voltage (over the LED Vf) across itself.

When a resistor is used on a common cathode, the current through this resistor would vary by the number of segments lit at any time. Thus, the voltage dropped would change as well. This will lead to intensity of digits changing with number of lit segments.