In this work we study 35 stellar clusters in the Small Magellanic Cloud (SMC) in order to provide their mean metallicities and ages. We also provide mean metallicities of the fields surrounding the clusters. We used Stroemgren photometry obtained with the 4.1 m SOAR telescope and take advantage of (b-y) and m1 colors for which there is a metallicity calibration presented in the literature. The spatial metallicity and age distributions of clusters across the SMC are investigated using the results obtained by Stroemgren photometry. We confirm earlier observations that younger, more metal-rich star clusters are concentrated in the central regions of the galaxy, while older, more metal-poor clusters are located farther from the SMC center. We construct the age-metallicity relation for the studied clusters and find good agreement with theoretical models of chemical enrichment, and with other literature age and metallicity values for those clusters. We also provide the mean metallicities for old and young populations of the field stars surrounding the clusters, and find the latter to be in good agreement with recent studies of the SMC Cepheid population. Finally, the Stroemgren photometry obtained for this study is made publicly available.