The first thing that scientists will learn, is that there is a severely blueshifted object near Jupiter. This will be extremely bright for the same reason that a sonic boom is loud. This "luminal boom" will not be noticed by most people on Earth, though.
The blueshift will be far outside of the visible range, well past X-rays, and into Gamma rays. The broad-spectrum gamma ray observatories -- satellites that are tasked with detecting the Gamma Ray Bursts associated with hypernovas, and the double-spikes of gamma radiation associated with nuclear weapons testing -- will be the first to detect the blueshifted light from our FTL ship as it decelerates through the transluminal barrier.
In minutes, the directional gamma ray observatories will be retasked to observe the extremely strange signal... Extremely strange, because it's the only gamma ray source that isn't just a huge, short lived spike of high energy photons at some arbitrary wavelength, but is a long lasting source that is changing its wavelength very, very, very quickly.
If they're lucky, they'll notice the streak of very high energy gamma radiation traveling away from the point where the FTL ship crossed the transluminal barrier, but this signal will likely be completely lost in the huge amount of gamma -- then X-ray photons that the FTL ship is reflecting.
As the ship slows down, we'll be able to better resolve it -- both because it's closer, and because our abilities to focus photons improves as the frequency goes down. It won't be in visible wavelengths until it's at least .25c, though.
Our scientists will know that something is very strange. Some will suggest FTL. Most will say that FTL shouldn't be possible. Unless we notice (and understand) the stream of gamma radiation that's traveling away from the point where the ship crossed the transluminal barrier, though, there will be no proof that the ship ever travelled faster than light, since simply travelling near the speed of light, then decelerating as it nears Jupiter will have the same effect.
Even if instruments do record that telltale FTL echo, it isn't conclusive proof, as the same signal could be produced from a high energy communications pulse being reflected by interplanetary dust, perhaps letting their homeworld know that they've arrived.
As the ship starts to depart our system, it will be severely redshifted, quickly becoming undetectable in the visible spectrum as the photons elongate out to radio frequencies. If it happened to have a clock on the rear, we would notice the clock slowing. Even though the redshift will happen at the same rate that we'd expect a ship traveling at near the speed of light will happen, the brightness will drop faster than we'd expect. This will be another hotly debated but still inconclusive detail in academia.
In all, several people will be convinced that FTL is possible -- a minority of physicists will deeply consider the implications but, unless they can replicate it, the only papers in journals will be theoretical discussions that conclude that more research is necessary.
However, seeing the object accelerate from (presumably) .999g to a relative standstill in the course of an hour, then back up to speed again over the next hour will generate a LOT of buzz. Physics will be turned over. I can't even begin to imagine the technology it would take to shove that much energy into a spacecraft in such a short amount of time without it turning into a stream of highly ionized and definitely-not-spacecraft-shaped dust. Cats-and-dogs-living-together scales of mass hysteria.