  Josephson scanning tunneling microscopy (JSTM) is a powerful probe of the
local superconducting order parameter, but studies have been largely limited to
cases where superconducting sample and superconducting tip both have the same
gap symmetry -- either s-wave or d-wave. It has been generally assumed that in
an ideal $s$-to-$d$ JSTM experiment the critical current would vanish
everywhere, as expected for ideal $c$-axis planar junctions. We show here that
this is not the case. Employing first-principles Wannier functions for
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$, we develop a scheme to compute Josephson
critical current ($I_{c}$) and quasiparticle tunneling current measured by JSTM
with sub-angstrom resolution. We demonstrate that the critical current for
tunneling between an s-wave tip and a superconducting cuprate sample has
largest magnitude above O sites and it vanishes above Cu sites. $I_{c}$ changes
sign under $\pi/2$-rotation and its average over a unit cell vanishes, as a
direct consequence of the $d$-wave gap symmetry in cuprates. Further, we show
that $I_{c}$ is strongly suppressed in the close vicinity of a Zn-like impurity
owing to suppression of the superconducting order parameter. More
interestingly, $I_{c}$ acquires non-vanishing values above the Cu sites near
the impurity. The critical current modulations produced by the impurity occur
at characteristic wavevectors distinct from the quasiparticle interference
(QPI) analogue. Furthermore, the quasiparticle tunneling spectra in the JSTM
set-up shows coherence peaks and impurity-induced resonances shifted by the
$s$-wave tip gap. We discuss similarities and differences in JSTM observables
and conventional STM observables, making specific predictions that can be
tested in future JSTM experiments.
