Rome has the same problems with Ciampino airport. It's exclusively used for budget airlines, and although it's closer to the city center there is no public transit link. You can either take the privately run air coach which takes forever (if the bus departs on time it should take 40 minutes, but late at night or early morning it's usually late taking double that or more) or cough up for a taxi and hope they don't rip you off (the fixed price only applies within the historic city walls, if you are going outside they will charge you more, even if it's less distance).

When I was living there I'd always try to fly to Fiumicino because of that. I can't imagine what impression it must give to tourists on their first trip to the city.