We consider planar stationary exponential Last Passage Percolation in the positive quadrant with boundary weights. For $ρ\in (0,1)$ and points $v_N=((1-ρ)^2 N,ρ^2 N)$ going to infinity along the characteristic direction, we establish right tail estimates with the optimal exponent for the exit time of the geodesic, along with optimal exponent estimates for the upper tail moderate deviations for the passage time. For the case $ρ=\frac{1}{2}$ in the stationary model, we establish the lower bound estimate with the optimal exponent for the lower tail of the passage time. Our arguments are based on moderate deviation estimates for point-to-point and point-to-line exponential Last Passage Percolation which are obtained via random matrix estimates.