Last summer I did a 3,400 round trip to Italy via Switzerland with no issues so I can certainly recommend a long road trip in a Fulvia!
I agree with the other folk who have replied on the basis that prevention is better than cure. I used the time before I went to clear up some messy wiring relating to the ignition and other minor niggles to de-risk any issues as well as doing full fluid changes (and I can thoroughly recommend an addition like Millers Extra Cool). I also took the Fulvia to a Lancia specialist for a tune as well as a full check over for extra peace of mind.
Do you normally have your fuel tank fully filled? I hadn't normally as with shorter journeys and advice that petrol 'goes off' I would only have a half tank of petrol at any time. It was only shortly before my long trip that I filled the fuel tank fully to find that the seal between filler pipe and tank had perished allowing fuel to leak; something that would have been a PITA to fix on the journey and not something I had predicted or even thought about. Fortunately finding this in advance meant it could be fixed.
With all this prep work I then took only a fairly basic tool kit and spares. The only thing I wish I'd taken? ... some copper grease! The brakes developed a squeal on one side at the rear (probably all the brake activity on those Swiss mountain passes!) and where a coating of some copper grease on the back of the offending pads would cure this niggle. I must have visited about a half dozen petrol stations en-route to try and find some with no luck and couldn't find any Swiss equivalents of Halfords! In the end I used the local knowledge of some chap admiring the Fulvia to locate a friendly back street mechanic and blagged some grease from him!
For additional peace of mind it's also worth considering breakdown cover and on that check your classic insurance policy. My policy for instance handily includes UK and European breakdown assistance.