Running climate simulations informs us of future climate change. However, it is computationally expensive to resolve complex climate processes numerically. As one way to speed up climate simulations, neural networks have been used to downscale climate variables from fast-running low-resolution simulations. So far, all neural network downscaling models can only downscale input samples with a pre-defined upsampling factor. In this work, we propose a Fourier neural operator downscaling model. It trains with data of a small upsampling factor and then can zero-shot downscale its input to arbitrary unseen high-resolutions. Evaluated on Navier-Stokes equation solution data and ERA5 water content data, our downscaling model demonstrates better performance than widely used convolutional and adversarial generative super-resolution models in both learned and zero-shot downscaling. Our model's performance is further boosted when a constraint layer is applied. In the end, we show that by combining our downscaling model with a low-resolution numerical PDE solver, the downscaled solution outperforms the solution of the state-of-the-art high-resolution data-driven solver. Our model can be used to cheaply and accurately generate arbitrarily high-resolution climate simulation data with fast-running low-resolution simulation as input.