Just keep the input levels to the Arduino under 5V and you will be fine. The high and low voltage circuits can of course share a ground if electrically necessary.
For outputs to high voltage circuitry, the safe way to do it is with an optoisolator; read this for more info. With an optoisolator, your low voltage (Arduino) sends signals optically to the high voltage Nixies, with no chance of the HV or associated noise coming back down the line.