The time-spectral CFD method (TSM) promises a huge efficiency potential for engineering flows in which the solution can be expected to be predominantly periodic in time, such as engine, wind turbine, helicopter or valve flow. For N harmonics in time, the TSM leads to a nonlinear equation system in the time domain governing a blocked state vector composed of 2 N + 1 steady-state like CFD solution vector blocks. Each vector block is associated with a collocation point in time. Standard spatial discretization schemes can be used for the individual collocation points that are coupled in time via the TSM time operator. Such coupled, large nonlinear block equation systems require scalable and robust implicit solution algorithms to leverage the efficiency potential of the TSM for complex engineering applications. The TSM was integrated into the compressible CFD solver CODA, developed as part of a collaboration between the French Aerospace Lab ONERA, the German Aerospace Center (DLR), Airbus, and their European research partners. We will discuss key aspects of the TSM integration into the implicit CFD software CODA for unstructured grids. The focus is on the extension and the tailoring of the implicit solution method for TSM equation systems. The approach relies on a CFL-relaxed pseudo Newton-algorithm, in which the linear equation systems can be solved by nested solver stacks, such as Krylov/GMRES schemes which, in turn, can be preconditioned by different methods such as Block-Gauss-Seidel/Jacobi with direct element-wise block inversion based on the LU method. Moreover, line-inversions based on the Thomas algorithm and taken from the base-line solver can be reused in the solver stack for the TSM problems. Different solver settings for the hierarchical solution method will be investigated and discussed in terms of robustness and scalability for different aeronautic configurations. Among the targeted, viscous aerodynamic simulation cases are fully turbulent 2D wing sections or 3D wing/wing-body configurations, for which the TSM method will be studied in combination with both standard 2nd-order finite-volume (FV) and higher-order discontinuous-Galerkin (DG) schemes in space. A key ingredient to the presented solution approach are exact sensitivity derivatives of the residual computed by algorithmic differentiation (AD). The TSM will be verified against reference solutions, e.g. generated from implicit time-stepping methods.