score:3
This is unfortunately not quite clear.
The actual Schengen legislation that governs visa-free short stays does not clearly say that you must exit and re-enter the Schengen area in order to "trigger" your visa-free period. In fact it seems to be hard to argue that this is even a plausible interpretation of any text in the regulation.
On the other hand, there are frequent anecdotal report of people being told before their long-stay visa expires by embassy employees, border guards, or police that they should make such a "visa run" when they switch status.
But, to the best of my memory, we have never heard of anyone actually getting in trouble afterwards for not having done the visa run.
Since nobody seems to be actually getting in trouble, the European Court of Justice has likely not had any occasion to clarify what the regulation properly means, so the current lack of certainty may continue indefinitely.
At the end of the day it will be up to you yourself if you want to depend on what the rules seem to say, or make a detour on your way to Italy to reduce the risk of running into officials who understand the rules differently.
The relevant legislation is article 6, section 2, of the Schengen Borders Code:
For intended stays on the territory of the Member States of a duration of no more than 90 days in any 180-day period, [... ... ...]
For the purposes of implementing paragraph 1, the date of entry shall be considered as the first day of stay on the territory of the Member States and the date of exit shall be considered as the last day of stay on the territory of the Member States. Periods of stay authorised under a residence permit or a long-stay visa shall not be taken into account in the calculation of the duration of stay on the territory of the Member States.
(my bolding). It seems to be difficult to interpret this in other ways than to say that the parts of your travel plans that are not in Austria will have fallen under section 1 right from your initial entry because you don't have more than 90 days in Schengen that are not excluded from the count by the above section.