Urban-induced changes in local microclimate, such as the urban heat island effect and air pollution, are known to vary with city size,  leading to power law or logarithmic relations between average climate variables and city-scale quantities (e.g., total population or area). However, these approaches suffer from biases related to the choice of city boundaries and they neglect intra-urban variations of city properties. In this study we use high-resolution data of urban temperatures and annual concentrations of particulate matter together with population density and street network properties and show that their marginal and joint probability distributions follow universal finite-size scaling functions. These results extend previous findings on city-scale relations, offering a novel description of intra-urban fluctuations of climate characteristics.