Question: It has been said that our universe is going to eventually become a de Sitter universe. Expansion will accelerate until their relative speed become higher than the speed of light. 

So i want to understand what happens after this point: so from our point of view, we see a progressively shrinking event horizon (each galaxy sees one, arguably each spacetime point sees one). Now, what is the Hawking Radiation expected from this event horizon? It would seems that the event horizon area is shrinking around us, but its actually a black hole turns outwards: the black hole is actually ***outside*** the event horizon, and the visible "well-behaved" spacetime without singularities in the inside of this horizon

In any case, intuitively (i don't have any arguments to support this) i would expect that the Hawking Radiation *inside* the horizon would grew larger as this horizon would shrink, but i would love to hear what is actually expected to happen at this point

Answer: 
The Hawking radiation of a deSitter space is given, as always, from the near-horizon metric, consistently redshifted to fill all space. Normalizing the cosmological constant appropriately:

$$ ds^2 = - (1- {\Lambda\over 3} r^2) dt^2 + {dr^2 \over (1 - {\Lambda\over 3} r^2)} + r^2 d\Omega^2 $$

If you flip the sign of $dt$, you can identify the metric of a 4-sphere after appropriate coordinate transformations, and from this read off the periodicity of t, which goes around the sphere. But this is not necessary. The near horizon metric is Rindler (as usual for hot horizons) and writing $r=r\_0 - {6\over\Lambda} u^2 $ where $r\_0$ is the deSitter radius, you find:

$$ ds^2 = - ( {\Lambda^2 u^2\over 9}) dt^2 + du^2 $$

Which gives the imaginary time period is $\Lambda u\over 3 $, so that the near horizon temperature is $3\over\Lambda u$. Extending this using the redshift factor, the temperature at the center is

$$ 1\over {2\pi r\_0}$$

Where

$$ r\_0 = \sqrt{3\over\Lambda}$$

Which is the usual deSitter temperature. This temperature is locally the same everwhere, because the space is isotropic. The horizon is static, and stable in equilibrium with this thermal bath, it doesn't grow and it doesn't shrink.
Hawking radiation from a de Sitter horizon is the cosmic microwave background radiation. Both are black body radiation. The Big Bang is a myth.