Fate of Jupiter when our sun dies Five billion years from now, the sun will have grown into a red giant star, more than a hundred times larger than its current size.
  While this metamorphosis into the giant star will change the solar system, scientists are unsure what will happen to the third rock from the sun.We already know that our sun will be bigger and brighter, so that it will probably destroy any form of life on our planet. 
  But whether the Earth's rocky core will survive is uncertain.
  At the end of its evolution, seven billion years from now, the sun will become a tiny white dwarf star.


(by Leen Decin, professor at the KU Leuven Institute of Astronomy (source))

When our sun runs through its last phase of its life it will expand and tremendous solar flares will flow out of it. Earth will end up getting baked and will lose its atmosphere.

Jupiter is a gas giant planet, a GAS GIANT. So,

What will happen in Jupiter? Will it evolve into a star since it contains hydrogen and helium? How will the atmosphere deal with the extreme heat?
Jupiter won't evolve into a star, it is not big enough. A body would have to have about 80 times the mass of Jupiter for there to be significant fusion occurring in the core. The end of life of the Sun won't change the mass of Jupiter.

Jupiter will continue to orbit the Sun as it evolves into a red giant. Although the solar wind will be much much more powerful, it won't have a significant effect on the overall mass of Jupiter.

When the Sun loses its outer layers it will, in the last million years or so of its life, lose about half its mass. This will have a significant effect on the orbit of Jupiter, causing Jupiter to migrate outwards.  While it is possible that some planet's orbits may become unstable and get ejected from the solar system, it is more likely that Jupiter will settle into a new, wider orbit around the white dwarf.