  The superconducting symmetry of Sr$_2$RuO$_4$ remains a puzzle. Time-reversal
symmetry breaking $d_{x^2-y^2} + ig_{xy(x^2-y^2)}$ pairing has been proposed
for reconciling multiple key experiments. However, its stability remains
unclear. In this work, we theoretically study the superconducting instabilities
in Sr$_2$RuO$_4$, including the effects of spin-orbit coupling (SOC), in the
presence of both local and longer-range interactions within a random phase
approximation. We show that the inclusion of second nearest neighbor
repulsions, together with non-local SOC in the $B_{2g}$ channel or
orbital-anisotropy of the non-local interactions, can have a significant impact
on the stability of both $d_{x^2-y^2}$- and $g$-wave pairing channels. We
analyze the properties, such as Knight shift and spontaneous edge current, of
the realized $d_{x^2-y^2} + ig$, $s^{\prime}+id_{xy}$ and mixed helical
pairings in different parameter spaces and find that the $d_{x^2-y^2} + ig$
solution is in better agreement with the experimental data.
