I was arrested for Public Intoxication, if I do not show in court, do they have to nullify the claim?

Obviously no! If you do not show up in court, a warrant for your arrest will be taken out against you. You know you have a court date, which you acknowledged that you received when you posted bond. As such, you have been given notice of your required appearance in court. Plus, you have a contract with the bondsman to appear in court as notified. Additionally, you are now required to defend yourself against the charge.