We demonstrate single-shot imaging and narrow-line cooling of individual alkaline-earth atoms in optical tweezers; specifically, strontium trapped in 515.2-nm light. Our approach enables high-fidelity detection of single atoms by imaging photons from the broad singlet transition while cooling on the narrow intercombination line, and we extend this technique to highly uniform two-dimensional tweezer arrays with 121 sites. Cooling during imaging is based on a previously unobserved narrow-line Sisyphus mechanism, which we predict to be applicable in a wide variety of experimental situations. Further, we demonstrate optically resolved sideband cooling of a single atom to near the motional ground state of a tweezer, which is tuned to a magic-trapping configuration achieved by elliptical polarization. Finally, we present calculations, in agreement with our experimental results, that predict a linear-polarization and polarization-independent magic crossing at 520(2) nm and 500.65(50) nm, respectively. Our results pave the way for a wide range of novel experimental avenues based on individually controlled alkaline-earth atoms in tweezers-from fundamental experiments in atomic physics to quantum computing, simulation, and metrology.