Using a Green’s-function formalism, we study the spin-resolved local spectral function of a resonant impurity coupled to a two-dimensional $d$% -wave altermagnetic substrate. It is found that the interplay between direct tunneling from the impurity to the scanning tunneling microscopy (STM) tip and substrate-mediated tunneling gives rise to a dual Fano resonance in the absence of an external magnetic field. Moreover, the anisotropic spin-dependent oscillations of the local spectral function and the corresponding Fano factors provide information on the altermagnetic splitting strength from complementary local and global perspectives. In addition, spin-selective tunneling can be achieved by tuning the Fermi energy and the tip position. Under a Landau-quantizing magnetic field, altermagnetism distorts the spin-resolved nodal structures differently in the two spin channels, producing a mismatched nodal pattern and a large spin contrast in scanning tunneling spectroscopy (STS). These results establish resonant-impurity STM/STS as a phase-sensitive local probe of altermagnetic band anisotropy.