You are confusing two separate issues. One is jurisdiction and the other is lack of service.
You do not have to be a resident of NY for the court to have jurisdiction. The question is whether the debt is related to NY in any way such as the signing of the contract or the furnishing of services in NY.
Challenging for lack of service means that you were never notified of the court proceedings but that you agree that the court has jurisdiction.
You need to get a copy of the file from the court. The plaintiff has to show that there was service so you need to see how he satisfied that requirement. He also has to show the court has jurisdiction. You need all that information so you will know how to proceed.
If it is jurisdiction and you have no contacts with NY you probably want to wait to challenge jurisdiction until such time as they domesticate the judgment in your home state but that is a strategic decision.
You just file a motion to set aside the judgment and state your reason. You will follow the heading and use the other court filings as a guide or the clerk of court may print out a motion from a similar case for you to use as a reference.
This is not from NY but should give you an idea
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