There might be a simpler way, but here’s what I’d do:
Put a bucket under the trap.
Underneath the sink, there’s a nut that holds the pivot bar into the drain and through the stopper; if your drain-pipe is plastic, the nut should be plastic, also–probably a wing-nut. Twist the nut off the D-P. Pull the pivot bar out of the stopper/D-P (there’s a plastic ball on the pivot bar that seats into the D-P). Remove the stopper. You might be able to see gasket material blocking the overflow exit. Pour water into the overflow and see if it drains at about the same rate. If so, reassemble in reverse order; your problem is somewhere else. If the water doesn’t drain through the overflow sufficiently, and/or you can see the gasket material blocking the overflow drain, I’d call a plumber (that’s just me; a novice plumber can remove the upper drain pipe and fix all that, but I wouldn’t, especially if, like me, you’re renting.) Replacing the pivot bar into the stopper can be a real bastard, just have patience and persevere.
IANAPlumber; any/all comments encouraged.
*While you have the stopper out, shine a flashlight down the drain and see how junked up it is. If it’s grotty, get a bottle brush and scrub. Flush with very hot water.