No, you don't need to board the train to see the island: it's rushed way to take you through some of the most interesting sites. I only don't know why the NP office tries to pack most of visitors to Brijuni - if you don't want to go by it you're not entiteled to any cheaper entry ticket nor discount for renting a bike which is shame.
They also don't stop in safari nor by the olive tree and pine alley. They stop only by the elephant and for a quick snack there.
I don't know where exactly its route goes and heard it takes only one hr riding altogether... for that many interesting sites it's really not much.
Beside you don't have best views and it's cowded.
Unique Suggestions: Well you don't get tired because you don't walk. And maybe you don't get hugry either.
Fun Alternatives: Walk or cycle. Enjoy the pleasure of unspoiled air and there's no lot noise from cars or moped.
Well I guess this tip may useful to some who're not loaded with cash: because Brioni tries to be 'exclusive' destination catering to boutique tourism most of hotels and restaurants belong to upper scale. I don't mean to encourage you to avoid them altogether but just warn that you'll pay more than in those by the continental eateries.
It makes lot of sense of course.
Unique Suggestions: There are no shops on island to buy food so bring your own sandwich from Fazana. Well, if you want to eat then try nice restaurant by the hotel Neptun next to port with some niceviews and classy atmosphere. Actually that's really nice place to sit down after whole day out doing activites.