The route via bar-and even more via ALB, will eat at last 2 days out of that 4 so not lve that much time for beach and sunny. Plus that coast near Bar is not that pretty.

SPLIT is the usual ferryport-and as most ferries run overnite to Ancona, that is more handy. But rhen Anc->Nap is not that str8 forward-via Roma and then Frecce.

Belgrade-Bar and then the ferry is definitely the most scenic way.
But you can also go via Albania after Bar (Durres-Ancona e.g.) or Split-Ancona. 
In summer there is usually several times per week a nighttrain from Budapest to Split and that will be the most relaxing way if you have only 4 days. 
Keep in account that the train(s) to Split are often delayed in summer (but maybe this year it gets better as no corona and Croatia is finally a Schengen member) - the line to Split is single-track only...

There are some ferries from Croatia to Italy, if that's what you mean. https://www.interrail.eu/en/plan-your-trip/tips-and-tricks/trains-europe/ferries/snav-ferries

I myself want to take the train over the Belgrade-Bar (Montenegro) route sometime - I've heard it's pretty scenic and I've never been to Serbia nor Montenegro - and there seem to be ferries from there to Bari in summer, but unfortunately for the time being there are no trains from Budapest to Belgrade.