You have a common cathode display, & NPN transistors or ULN2003/ULN2803?
Shift data into the shift register, turn on cathode 0.
2mS later, turn off cathode
Shift data into the shift register, turn on cathode 1.
2mS later, turn off cathode
Shift data into the shift register, turn on cathode 2.
2mS later, turn off cathode
Shift data into the shift register, turn on cathode 3.
2mS later, turn off cathode
During the 2mS, do other stuff.
This is blink without delay style coding.
