Modern gas turbines find extensive applications in aero engines, power generation, marine, and various other industries. Most numerical studies concentrate on turbine aero-thermal performance under different external conditions with vane or coupon; there are a few published results on combustor-turbine interactions. This study reveals the cooling performance of the first stage with or without coatings to provide a reference for high-performance combustors and turbine-integrated design. The results of the study show that (1) The velocity and temperature distribution inside the combustion chamber are obviously affected by the swirling flow. A central recirculation zone is formed near the central axis, and two external recirculation zones are formed between the inlet section of the combustion chamber and the fluid reattachment point. (2) Inside the combustion chamber, the flame temperature in the central recirculation zone is relatively high, and the range of the high-temperature zone expands with the increase of axial distance. Increasing the swirl number decreases the peak temperature level in the combustion chamber. (3) Under the influence of swirl number, the greater the swirl number, the greater the cooling effectiveness of most areas on the vane surface. (4) In regions where there is a decrease in local heat flux, positive values are evident. This suggests that the application of a coating in these areas results in a reduction of heat transfer from the vane to the mainstream. (5) When comparing the coated vane to the uncoated vane, the cooling effectiveness across the entire surface is notably enhanced, with a particularly significant improvement observed on the vane’s suction side. With the increase of Cax, the difference in cooling effectiveness increment under different swirl numbers also increases.