Just want to make sure I've got this right.

If I use Premonition to reveal and use a card set aside, what happen to this card if I can't resolve it? Is it still discarded or does it stays in the set aside zone with the Premonition card?

Let's say I have only Grievous (red character) left. I use Premonitions to reveal a 'Rise Again' card on the set aside card. Rise Again only works on unique blue character so I can't resolve it. What happens next?

Premonitions plays the card for free, so you play the card, but it does nothing.

In your example, you play Rise Again for free, can't spot a character to heal damage from, heal no damage, and then don't play the free card from Rise Again because it says "then" and you couldn't finish the first part.

It doesn't say "then".

The full text reads...

"Heal 5 damage from a unique Blue character. You may play an upgrade from your discard pile on it for free."

But you are still right. You cannot play the free upgrade because it has to go on the unique Blue character that was healed (defined by the "on it" text), which there isn't one of to heal.

My question is, what happens to the 'Rise Again' card if you can't play it? Does it stay on 'Premonition' in the set aside zone or is it discarded?

You can play Rise Again, you just can't use it. So it's effectively discarded. It does, however, trigger any effects that would trigger when you "play" an event (i.e. Mon Mothma's ability)

If there are no unique blue characters in play, it gets played and does nothing upon resolving.

If your opponent has a unique blue character, you'd have to heal it, but could then choose not play an upgrade on it.