I went for work earlier this year. Went out early so I could spend a day mooching about. Should have hired a car, I probably spent the same in taxis to and from the airport. Plenty to do for a day without one, though.
There’s a special bus that runs from town to the bottom of Puy de Dome. Various walks to the top (including some beautiful, sinewy, steep singletrack through the woods that had me aching for a bike!) – much nicer than taking the funicular. 90 mins give or take, depending on the route. Can’t get lost – keep going up! Great views from the top. You could walk a seciton of GR4 if that sort of thing floats your boat – it takes in the summit of P de D
The town’s nice, lots of places to eat and drink, pavement cafes in the old town, everything you’d expect.