mal,
Reservations are provided with any tickets you buy, if they're required on the particular train you're using. That's the case whether you buy them from the website you linked, from Trenitalia (the operator of the rail system) or from other resellers such as www.capiatainetrain.com. You don't "need" to make reservations (buy tickets) far in advance, however some advance purchase tickets are heavily discounted (depending on how far in advance they're purchased) so it's always good to save money.
One of the easiest and quickest trips from Orvieto to Napoli Centrale is a departure 11:23, arriving Napoli Centrale at 14:15 (time 2H:52M, one change at Roma Termini). Current fare is listed at €31.50 PP (the second segment is via Freccia high speed, and there are still Super Economy tickets available for that date - however note that Super Economy are non-refundable and non-changeable so choose carefully). You can easily buy tickets for the first segment (Regionale) when in Italy as there's no advantage to buying in advance. DON'T forget to validate Regionale tickets prior to boarding the train on the day of travel, or you'll risk hefty fines!
I can't comment on whether the site is "accurate" as I'm not clear on which particular aspect of the site you're asking about. It's a well known website, and I'm sure anything they have listed will be accurate.