The outer heliosphere is an interesting region characterized by the interaction between the solar wind and the interstellar neutral atoms. Having accomplished the mission to Pluto in 2015 and currently on the way to the Kuiper Belt, the New Horizons spacecraft is following the footsteps of the two Voyager spacecraft that previously explored this region lying roughly beyond 30 AU from the Sun. We model the three-dimensional, time-dependent solar wind plasma flow to the outer heliosphere using our own software Multi-Scale Fluid-Kinetic Simulation Suite (MS-FLUKSS), which, in addition to the thermal solar wind plasma, takes into account charge exchange of the solar wind protons with interstellar neutral atoms and treats nonthermal ions (i.e., pickup ions) born during this process as a separate fluid. Additionally, MS-FLUKSS allows us to model turbulence generated by pickup ions. We use MS-FLUKSS to investigate the evolution of plasma and turbulent fluctuations along the trajectory of the New Horizons spacecraft using plasma and turbulence parameters from OMNI data as time-dependent boundary conditions at 1 AU for the Reynolds-averaged MHD equations. We compare the model with in situ plasma observations by New Horizons, Voyager 2, and Ulysses. We also compare the model pickup proton parameters with those derived from the Ulysses-SWICS data.