If I were in your character's situation, I imagine it would fire the graple at the edge of the station, the "horizon," so to speak. I would not fire at the edge that is moving toward me, but the opposite edge. That way, as my grapple made contact with the surface, it would start out in a motion almost directly away from me, with very little sideways motion. As the grapple disappears over the horizon, all of the "pulling" force that I experience would come from that horizon; the exact point where the cable makes contact with the side of the station.
Now, if I had any choice, I would latch on to the side of the station the instant I came within reach of it. Then, I would experience the same 1g as someone standing against the hull on the inside of the station. However, I would experience it in the opposite direction, as though I were hanging from a ceiling with the "bottomless drop" of outerspace beneath me. This would be a terribly uncomfortable situation, and I would want some sort of hook or clamping mechanism so that I could attach myself to the hull (rather than trying to hang on with my own strength).
To add to the drama, if your character misses his opportunity to attach to the bulkhead on the first Pass, he will overshoot and be flung back out into space. Once he passes the edge of the station which once represented its horizon to him, his cable will begin to pull with an angular motion. He would be flung out to the full length of the cable (much like a tetherball with its rope unwinding). Once at the end of the cable, he would "hang" from it while experiencing a centrifugal force of significantly more than 1 g. The amount of this force would depend on the length of the cable; as your hero gets further from the center of rotation, but the period of rotation remains constant, the centrifugal force increases.
