High-resolution angle- and spin-resolved photoemission spectroscopy reveals new features of the electronic structure of the ferromagnetic triple-layered ruthenate Sr$_4$Ru$_3$O$_{10}$. There are narrow spectral peaks $\sim$30 meV below the Fermi-level in two regions of the Brillouin zone: a hole-like band at the zone-center and a saddle-point van Hove singularity at the zone edge. Below T$_c$ each feature is almost completely spin-polarized, with opposite sign polarization and with strong Kondo-coherence-like temperature-dependent spectral weight variation suggestive of strong Hund's metal correlations. In addition, there are distinct Fermi surfaces for wide electron-like minority spin bands around the zone center and narrow hole-like majority spin Fermi surface contours around the zone corners. The origin of these features is in general agreement with density functional calculations, if they are shifted to reduce the exchange splitting, and scaled to take into account effects of correlation. Furthermore, the deduced narrow band origins from the tri-layer splitting of $d_{xz/yz}$ orbitals implies a layer-specific spin-polarization contribution from the narrow bands. Over a larger energy range, net spin-majority polarization of incoherent Ru $d$-bands is observed to extend down to the top of the oxygen bands, where additional narrow oxygen bands contribute to the magnetism with spin-minority polarization.