  Astronomical tests of spacetime metric and gravitation theory near the
Galactic Center (GC) black hole, Sgr A* have gained momentum with the
observations of compact stellar orbits near the black hole and measurement of
the black hole shadow. Deviation from the Kerr metric is a potential signature
of modified gravity theory. In this work, we use Newman-Janis algorithm to
construct an axially symmetric and asymptotically flat metric in f(R) scalaron
gravity theory. We call it as Kerr-scalaron metric. For studying astronomical
consequences of the new metric we use the compact stellar orbits and the black
hole shadow. We use the observed size of the emission ring of the GC black hole
shadow for estimating deviation of the new metric from general relativity. It
has been found that scalarons with mass within $10^{-17}$ eV - $10^{-16}$ eV
are compatible with the observed emission ring size for black hole spin
$\chi=0.9$. Schwarzschild limit of the pericenter shift is estimated for
compact stellar orbits near the black hole. General relativistic pericenter
shift in wider orbits including S-stars such as S4716 and S2 has been
reproduced with these scalarons. The parameter $f_{SP}$ measuring deviation
from Schwarzschild pericenter shift has been found as $f_{SP}=1.00-1.04$ within
stellar orbits having semi-major axes $45$ au - $100$ au. Scalarons have the
capability to dominate Schwarzschild precession for orbits much below $45$ au.
Lense-Thirring (LT) precession with the new metric is estimated for the compact
orbits. The massive scalarons produce LT precession with magnitude
($12.25-24.5$) $\mu$as/yr in the orbit of S2. The LT precession time scale is
within $0.1$% of the age of the S-stars.
