How long between Arduino digital pin and IC?
The 74HC595 is CMOS technology so it should take almost nothing in the way of current to drive it, so IR drop wouldn't be a concern. 

As long as you keep the frequency of the signals down below say 100kHz, you shouldn't have to worry about transmission line effects. Assuming your intended observer for the LEDs to be the human eye, you shouldn't have to worry about high speeds anyway. For example, 8 digits at 7 segments and a decimal point each is 64 LED elements, and at a mere 9600 bps, you could update the display in just under 7 msec. 

The only thing I would worry about is whether the digital high output level from the Arduino will register as a high input at your shift register. As long as the shift register is running from a 5V supply (and not something odd like 6) you should be fine there, too. (and if this was going to be a problem, it would manifest itself over a mere 10cm of wire so that's easy to check)

Short answer: very high probability you can go from arduino to cable to 74HC595s no problem.