Poipu is about 15 miles from Lihue Airport. The drive usually takes 30 to 40 minutes via Highway 50 and Koloa Road, longer when south-shore traffic backs up near Koloa.
A rental car is the practical choice on Kauai. Taxis and rideshares can work for airport transfers, but they are a poor substitute for a car if you plan to visit Waimea Canyon, Hanalei, or trailheads around the island.