What happens if you rear end someone and there was very little damage and the accident was not reported to the insurance company or police dept can the other party still press charges one month after?

The other party can still file a claim with your insurance company, but it is highly unlikely that the Police would get involved at this point. Depending on what state you are in, the other party has years to file that claim. If the accident was that minor and they haven't reported it by now, they probably never will.