Considered here is the first initial boundary value problem for the 2D non-autonomous $g$-Navier-Stokes equations in bounded domains. We prove the existence of a pullback attractor in $V_g$ for the continuous process generated by strong solutions to the problem. We also prove the exponential growth in $V_g$ and in $H^2(\Omega ,g)$ for the pullback attractor, when time goes to $-\infty $.