So I will take a look at Giro d'Italia when in Italy in May. In one case (stage 3) it would be possible to first see the start in Rapallo at 13:35 and then move on to Barbagelata and see a mountain pass passing at ~16:20. According to google I will need some 60 minutes to drive there so will be there before 15 at least. But will the road close earlier?
Roads that the race actually follows are closed to cars a couple of hours before the racers arrive, sometimes more depending on the area. Cross roads are open until closer to race arrival time. Best bet is to drive nearby and get your bike out and ride the rest of the way to your preferred viewing point. Bikes and pedestrians can move along the closed course until the first riders are just a few kilometers away.
Finding a spot near the start can also be challenging, as a lot of the start area is reserved for VIPs, journalists, etc.