  The discovery of radio pulsars in compact orbits around Sgr A* would allow an
unprecedented and detailed investigation of the spacetime of the supermassive
black hole. This paper shows that pulsar timing, including that of a single
pulsar, has the potential to provide novel tests of general relativity, in
particular its cosmic censorship conjecture and no-hair theorem for rotating
black holes. These experiments can be performed by timing observations with 100
micro-second precision, achievable with the Square Kilometre Array for a normal
pulsar at frequency above 15 GHz. Based on the standard pulsar timing
technique, we develop a method that allows the determination of the mass, spin,
and quadrupole moment of Sgr A*, and provides a consistent covariance analysis
of the measurement errors. Furthermore, we test this method in detailed mock
data simulations. It seems likely that only for orbital periods below ~0.3 yr
is there the possibility of having negligible external perturbations. For such
orbits we expect a ~10^-3 test of the frame dragging and a ~10^-2 test of the
no-hair theorem within 5 years, if Sgr A* is spinning rapidly. Our method is
also capable of identifying perturbations caused by distributed mass around Sgr
A*, thus providing high confidence in these gravity tests. Our analysis is not
affected by uncertainties in our knowledge of the distance to the Galactic
center, R0. A combination of pulsar timing with the astrometric results of
stellar orbits would greatly improve the measurement precision of R0.
