I filed an appeal in the Judicial Court and unknowingly missed the appeal filing deadline. The judge dismissed my case.

What kind of case was this? From what court did you appeal and in what court did the judge rule that you too late?

It is very unclear in what court you are in or what the actual requirement is that you are asking about. Depending on the forum, the date for filing a notice of appeal may be jurisdictional -- meaning that if you miss that date, there may be no remedy. However, relief is often given for missing other appellate deadlines if you ask for relief soon enough. You do need to confer with an experienced attorney.

I agree with the other attorneys that you must specify what type of proceedings and appeal you are discussing. Your ability to reverse the untimeliness finding depends on the type of case you have and the type of appeal you were trying to pursue.