What is the ruling on shunning an ill-mannered person who does not pray and has wronged me and my relatives, and will one be punished for shunning him after greeting him in a gathering without shaking hands?

The basic principle is the impermissibility of a Muslim abandoning his Muslim brother for more than three days, unless there is a legitimate Shar'i reason that brings about a Shar'i benefit. For example, if the abandonment is due to bad manners, neglecting prayer, or harming others, then abandonment is permissible if it leads to a benefit and does not result in a greater harm. This should be done after advice has been given and proven ineffective. Furthermore, extending greetings (giving Salam) breaks the abandonment according to many scholars.