I would want to give the zombie a certain score in a scoreboard objective after getting hit by a specifically-tagged arrow. The closest I've figured out to doing this is detecting for the arrow nearby, then checking to see if the zombie has taken damage using the HurtTime datatag.
if its health, the score of the objective would be your health. Meaning, if your health is full, it will show a score of 20. So maybe you could set something up to detect whether the score is below or the same as 20. I'm not sure if this helps what you need but this is all I can do.