Yes, it'd be neat alright, but I hope I'm not here if/when it happens. Another star that close, even a tiny one, I'm pretty sure would render the earth uninhabitable. Doesn't seem likely anytime soon thought. Minimum size for a star--the point at which thermonuclear fusion would start--I think is about 4% of the solar mass, if my memory is correct, and Jupiter's well below that, only about 0.1% of the solar mass. It'd have to to be 40 times as big as it is now to ignite. Even if it somehow sucked up all the rest of the planets, all the gas and dust and asteroids and other small objects in the solar system, and all the Oort Cloud and Kuiper Belt objects, it'd be only 2 or 3 times its present size. The solar system would have to pass through a fairly dense interstellar cloud of gas and dust or encounter multiple pretty large objects for Jupiter to accrete that much mass.
Could happen though, in the very long term. The solar system not only rotates around the galactic core like everything else in the galaxy, with a period of 225 to 250 million years, it also, according to what I've been reading lately, oscillates vertically about the plane of the galaxy on about a 62 million year cycle. That cycle seems to be related to global mass extinction events. When we're on the north side of the galaxy (that's defined by the "right hand rule:" make a fist with your fingers curled in the galaxy's direction of rotation and your upraised thumb defines north) we're exposed to a greater cosmic ray flux because there's a great clump of galaxies called the Virgo Cluster on that side of our galaxy. When we're on the south side of our galaxy, there's the whole thickness of the galaxy between us and the Virgo Cluster, about 10,000 light years worth of gas and dust and other stuff that absorbs it. That relationship is not definitively established yet, all we have so far is some tantalizingly suggestive data...
But that's the kind of stuff I mean when I say that reality's far more interesting.