What is central to the question of Anarcho-Capitalism and Christianity is the role, source, and execution of law.
Christianity is wholly consistent with a Natural Rights understanding of the scope of law. It is also especially consistent with a Common Law process for derivation of law. Finally, Christianity is peculiarly inconsistent with a legislative process for derivation of law.
The only legitimate role of government from the Christian perspective is Judicial. The Judges were to read, interpret and, rule using the revealed law as delivered to Moses. The book of Deuteronomy consists primarily of the case law up to the time of Joshua.
The courts were to try cases, and anyone not willing to come under the ruling of the Judges was considered an outlaw, outside the protection of the law. The costs incurred in the judicial process were to be born by the individuals involved in the case.
Insomuch as anarcho-capitalism attempts to make the judicial a private function it may or may not be consistent with Christian thought.
Any other role of government beyond the judicial is wholly outside the scriptural mandate for the state.