I did a Monday-Saturday in Sardinia, we really did a marathon, starting out by ferry to Olbia, overnight at a marina resort right outside (cannot remember for the life of me what it was, but only 20 min outside of town), then drove across the island to Stintino for a night, then passed through Alghero on the way down to Pula to stay 2 nights to do some caves and such and Cagliari, and then back up to Olbia before the ferry back to the mainland.....it was crazy fast, and we didn't do any area much justice....though we did do some interesting things, like swimming to the island off of Stintino, driving into the mountains in the interior, and exploring Cagliari.....
I would suggest for 4 days, you could probably do Cagliari and the southern portion, or Alghero and the northern......definitely don't do the whole thing....
There are a lot of beach resorts in the north, so you might want to factor in that if you are hoping for some relaxation.....
Though I just saw that you would be in October.....I would probably base in a bigger city, maybe Alghero and Cagliari for half and half, because many of the smaller towns that cater to tourists might be a little on the dead side.....we went in April (arrived on Pasquetta/Easter Monday) and it was still quite dead....