Although the statute of limitations does say one year, there are other nuances which can add months to that deadline. You should bring all of your paperwork in to a lawyer who can help you sort through it. A valid statute of limitations violation is darn near an absolute defense.
The statute is one year for misdemeanor offenses in Arizona, but time is tolled (or NOT counted) if the case is on bench warrant status. If the Court mailed out a summons, and it was returned unclaimed (because you moved and left no forwarding address, for example), all of the time between when they sent it out and when you finally appear on it is NOT counted. The reasoning is, every accused criminal would try to avoid service of a summons on a misdemeanor complaint if they knew the case would be dismissed within a year.
As to whether a Justice Court is the proper forum, yes, a Justice Court is a court of limited jurisidition that has been empowered by the legislature and Supreme Court of Arizona to hear misdemeanor cases.