  Future observations with next-generation large-area radio telescopes are
expected to discover radio pulsars (PSRs) closely orbiting around
Sagittarius~A* (Sgr~A*), the supermassive black hole (SMBH) dwelling at our
Galactic Center (GC). Such a system can provide a unique laboratory for testing
General Relativity (GR), as well as the astrophysics around the GC. In this
paper, we provide a numerical timing model for PSR-SMBH systems based on the
post-Newtonian (PN) equation of motion, and use it to explore the prospects of
measuring the black hole (BH) properties with pulsar timing. We further
consider the perturbation caused by the dark matter (DM) distribution around
Sgr~A*, and the possibility of constraining DM models with PSR-SMBH systems.
Assuming a 5-year observation of a normal pulsar in an eccentric ($e=0.8$)
orbit with an orbital period $P_b = 0.5\,$yr, we find that -- with weekly
recorded times of arrival (TOAs) and a timing precision of 1 ms -- the
power-law index of DM density distribution near the GC can be constrained to
about 20%. Such a measurement is comparable to those measurements at the
Galactic length scale but can reveal small-scale properties of the DM.
