What will a future cold and "dead" Jupiter look like? I heard Jupiter is dying like getting smaller. For example, from Guillot et al. 2004 3: The Interior of Jupiter (also here):

(Jupiter)... is still contracting at the rate of ~3 cm per year while its interior cools by ~1 K per million year.

When will it completely die, how much smaller will it be then, and what will it look like?
As earlier answers indicate, Jupiter will slowly shrink until it is some 10-20% smaller with elements slowly settling depending on density and solubility.
One can calculate a rough Kelvin timescale for the ratio between binding energy and the blackbody power of the surface to estimate how quickly it cools down (this number will be nudged by the above considerations a bit, so it is an order of magnitude estimate):
$$\tau_{Kelvin}=\frac{3GM^2}{20\pi\sigma R^3 T^4}.$$
For Jupiter this timescale is 54 billion years.
As the universe cools eventually it becomes a ball of layered solid degenerate matter, with a frozen hydrogen crust. This will take longer than $\tau_{Kelvin}$ due to various mild heating processes like crystallization heating, tidal heating from the moons, possibly dark matter annihilation, and other low-energy processes that keep objects warmer than the CMB.