You need a car. Fayetteville itself is small enough to walk in twenty minutes end to end, but the National Park and gorge access points are spread across several miles of winding roads on both sides of the bridge.
The roads down into the gorge (Kaymoor Road, Fayette Station Road, the Nuttallburg approach) are narrow, often single-lane, and steep. They are scenic but slow — give yourself extra time. Fayette Station Road, the historic route under the bridge to the river, is a one-way scenic drive worth doing in itself; it takes about 45 minutes including stops.
Local rafting outfitters provide shuttles between their put-ins, take-outs, and their lodging. Park ranger-led shuttle programs sometimes operate on Bridge Day and during peak season.