If some person has be sentence in court can reopen the case and keep fighting the same case in court?

You are talking about a right to appeal. If convicted you can appeal against the conviction (because you believe you should not have been found guilty) or against the sentence (because you think it was too harsh). There is usually a time limit - it depends where you are, but usually an appeal must be lodged within 30 days or some similar period.

However, an appeal against a conviction can be done later (even many years later) - but only if significant new evidence comes to light which might have led to a different result had it been known at the time of the original trial.

No. Once someone is sentenced in court, the case cannot be reopened unless there is new information to consider.

Only if he or his attorney can prove that a mistake was made during sentencing, or if there some new evidence pertinent to the case that was not revealed prior to or during sentencing.