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The official site mentions a shuttle bus service, I also googled but the infos are not clear, so I hope to find some firsthand experience.

I have a flight with a stop in Fiumicino, I will arrive in Terminal 1 and the next departure is from Terminal 5, the transfer time is around 5 hours.

Is it correct that I should go to Terminal 3 first (using the internal transfer area) and after exit from Terminal 3 and take the shuttle to Terminal 5?

The terminal 5 has a waiting area with some shops or is just a terminal for departure? In your opinion is better to wait some hours inside Terminal 3 before going out to take the shuttle bus?

If you arrive at Terminal 1 on a through ticket, or have checked your luggage to final destination, or are travelling with hand luggage only, you should be able to do an airside transfer from terminal 1. Terminal 5 is only for security checks for departing passengers and after passing security, all airside facilities actually exist at Terminal 3, including departure gates, shops and airline lounges.

This is the relevant information for airside transfer at Fiumicino on ADR's site. Note that there is no mention of Terminal 5 since the departures will be from the Terminal 3 G & H gates.

Here is a relatively recent post on Flyertalk which sounds similar to your situation, i.e. arriving on a domestic Alitalia flight and transferring to a flight to the USA.

If you do need to collect luggage at Terminal 1, you would exit landside and take relatively short walk to Terminal 3, although there are additional shuttle buses that circulate from the long term parking areas to the terminals that you could take from Terminal 1 to Terminal 3 if walking is a problem.