If your Eurail ticket covers Italy, then it would definitely cover any travel within the country, but it probably would not be a good use of the ticket - generally they only pay off when you use them for long haul travel. I shouldn't imagine the cost of a ticket from Venice to Pordenone would justify using your Eurail ticket.
You can double check your Eurail pass to see which countries it covers, I'm not sure which ones cover Hungary. However, if yours doesn't include Hungary, you can use your Eurail pass up to the border of Austria, and then you just have to pay for your ticket to Budapest, which is a very short ride, and won't cost much. Within Hungary, fares are so cheap that it really wouldn't be worth using a pass unless you have extra days to burn that you can't possibly use anywhere else.