They’re not “refueling” the Sun but restarting it somehow. There’s no real detail in this–it’s basically handwaving to give impetus for the plot–but they also avoid a lot of nonsensical technobabble.
Regarding the film, it is one of the most visually stunning science fiction films since 2001: A Space Odyssey, and it sets a very appropriate, somber mood. It’s best to see this in the cinema; you’ll lose much of this on the small screen. The plot developments themselves are kind of dumb; basically, the ship is poorly designed for its mission, it doesn’t carry the necessary auxillary craft, lacks redundency in mission critical systems (airlocks, gardens, communications systems, protective shielding, flight computer, et cetera), and the failure of the previous mission (which is a critical plot development) is a poorly explained hash, at best. There are also a couple of significant orbital mechanics problems with the story, but that’s only of interest to nerds who sit around plotting capture and swing-by trajectories as a hobby.
Anyway, it’s not that the film is bad, necessarily–it doesn’t descend into a logical singularity that Event Horizon collapsed into–it’s just a grand opportunity to do a Kubrick-like film that is totally wasted by an increasingly random plot. It also has every appearance of last minute, studio-executive-tampering editing; whole plotlines seem to kind of collapse or be cut off on the last couple of reels as the film barrels toward a typical blockbuster ending. I’ve wasted worse two hours of my life on more dreckful wastes of celluloid. It’s not a shining endorsement, I know, but it could be worse, especially this summer.
Oh, and both Michelle Yeoh and Rose Byrne are radiant. I’d watch either of them drink tea for two hours. Well, maybe and hour and a half.