Can't open the image, but using HC595 with current limiting resistors to sink current from individual cathodes is fine.
For brighter output, use TPIC6B595 to sink current instead, can do full 20mA on each segment without worry, vs being limited to 8-9mA per segment due to HC595 70mA total current limit.
With HC595, shifting 0 in will turn a segment on.
With 6B595, shift a 1 in turns the output on to go low and turn a segment on.
That display has dual blue serial LEDs per segment and needs more than 5V to turn them on.
Thus Vs = 7.5V or 9V on the common anode, needs current limiting resistors, and TPIC6B595 to sink current thru it.
Forward Voltage Vf Per Seg. : 6.4V typical, 7.6V max
Forward Voltage Vf Per Dot : 3.2V typical, 3.8V max
need different resistors for dots than segments.
(Vs - Vf)/.02 = resistor
Also need A1/A2 wired in parallel with a resistor each, D1/D2 wired in parallel with a resistor each, and G1/G2 wired in parallel with a resistor each.
Typically the data you send out will be:
0bdpGFEDCBA
and wire up the shift register to match:
Q7 dp
Q6 G
Q5 F
Q4 E
Q3 D
Q2 C
Q1 B
Q0 A