Altena is a small town and most things of interest are reachable on foot, though the town's steep geography means that "on foot" often involves some climbing. The town centre and the Deutsches Drahtmuseum are in the valley floor; the castle is on the cliff above.
The easiest way to reach the castle is the Burgaufzug — a tunnel elevator system that runs from near the town centre up through the rock to the castle level. It operates during castle opening hours and saves a stiff climb on the forest path. The path itself is not difficult if you are reasonably fit and the weather is dry.
A car is useful for reaching the wider Sauerland area and for day trips, but is not needed for the town itself. Parking is available in the valley.