The geodesic equations of general relativity that describe motion of a test particle in Kerr spacetime are solved exactly including the contribution from the cosmological constant. By applying the exact solution for the precession of the point of closest approach for the orbit of the test particle around the Kerr field, we calculate the relativistic effect of periapsis advance for the observed orbits of S-stars in the central arcsecond of our galaxy, assuming that the galactic centre is a Kerr black hole, for various values of the Kerr parameter including those supported by recent observations. The observation of the predicted effects can provide an important test of the theory of general relativity at the strong field regime. In addition, we derive the exact solution of timelike non-spherical polar and non-polar orbits. Exact expressions for the periastron (periapsis) advance and frame dragging (Lense-Thirring) effect for a test particle in a polar, non-spherical orbit in the Kerr gravitational field are derived and applied for practical calculations around the galactic centre. We subsequently derive an analytical expression for the periapsis precession for the equatorial non-circular orbit of a test particle around a rotating mass whose surrounding curved spacetime geometry is described by the Kerr-de Sitter field.