In This Paper, We First Propose and Analyze a Steady State Dual-Porosity-Navier–Stokes Model, Which Describes Both Dual-Porosity Flow and Free Flow (Governed by Navier–Stokes Equation) Coupled through Four Interface Conditions, Including the Beavers–Joseph Interface Condition. Then We Propose a Domain Decomposition Method for Efficiently Solving Such a Large Complex System. Robin Boundary Conditions Are Used to Decouple the Dual-Porosity Equations from the Navier–Stokes Equations in the Coupled System. based on the Two Decoupled Sub-Problems, a Parallel Robin-Robin Domain Decomposition Method is Constructed and Then Discretized by Finite Elements. We Analyze the Convergence of the Domain Decomposition Method with the Finite Element Discretization and Investigate the Effect of Robin Parameters on the Convergence, Which Also Provide Instructions for How to Choose the Robin Parameters in Practice. Three Cases of Robin Parameters Are Studied, Including a Difficult Case Which Was Not Fully Addressed in the Literature, and the Optimal Geometric Convergence Rate is Obtained. Numerical Experiments Are Presented to Verify the Theoretical Conclusions, Illustrate How the Theory Can Provide Instructions on Choosing Robin Parameters, and Show the Features of the Proposed Model and Domain Decomposition Method.