Can you get insurance to pay for hit and run damage when you didn't get the license plate number or if you are at fault?

Well, it's not uncommon for the hit-and-run vehicle and driver to be unknown, so not knowing the license shouldn't be a factor. But I'm confused by the "if you are At Fault" part of your question. Do you mean that you did the hitting but the other driver ran? Doesn't really matter, I guess. If you have collision and comprehensive insurance, it should pay the claim. If you have only liability insurance, you don't get anything.

As long as you have collision coverage