While I don't practice in your state, here is my limited answer to your question.
Generally, if its a Misdemeanor assault, the injured party has between a year and two years to bring the complaint to the police or DA.
If its a Felony, it can usually go as long as five years.
However, it unlikely that you will be charged if the "victim" hasn't yet gone to the police. In fact, the longer it takes to go to the police, the less likely they will be to entertain it.
The person does not press charges, the State or City does. Depending on the seriousness of the crime there could be no limit to the time the State has to charge you. The person you punched has 3 years to file a civil claim for damages. This is separate and apart from any criminal charges.