This means that if theoretically you were to faster than time, time inside the spaceship(and actually; the whole spaceship itself) would go back.
It wouldn't go back(wards). It would go off in a direction perpendicular to the way we currently perceive time to flow.
(Ah, I see kaiyuu, and now at least one other in the intervening posts, has covered that. Still, I'll leave this in.)
At lightspeed(which is impossible to get), and according to current theories you would indeed be "magically" frozen in time.
Whatever speed you go, you experience
your time at 1s/s. The rest of the universe is going to look weird, though.
It's probably easier to explain what happens at
not quite light speed, of course, given that (without some physical tricks) it is, as you say, impossible to go SoL, at least (give or take an opinion or two) if you have any rest-mass. (Isn't it that if you
don't have a rest-mass then you can't do anything
other than go SoL? I forget whether that's been poo-pooed as an idea.
Anyway, things
appear to almost instantly arrive at where you are, if they are heading straight towards you (i.e. you are heading straight towards them) because the light they send out (extremely compressed, probably dangerous to look at) and the object themselves are going (from your perspective) at (nearly) the SoL. Assuming your survived contact (either some method of 'only moving the rest of the universe around you for a given distance), or they were ever so slightly obliquely travelling past you then you'd get a marginally less pronounced 'flash' effect. If you're just skirting them and keeping on going then you'd get a zoomed-down effect (during which everything that happens to them during your objective travel-time gets revealed to you) until there's a frozen instant where you receive everything un-shifted, then it zooms off into extremely long-shifted light, and a slowed down 'historic view', as you barely get any new information, very soon everything being so stretched out that you lose any ability to detect them (notwithstanding the inverse-square law on the apparent distance between you and them meaning you're half a galaxy away as well, mere 'instants' of your perception... or perhaps it's that the near-light-speed "sun" that is receding from you so fast that it's leaving behind less and less observable evidence of its onwards track.)
The big problem is not
coasting at the speeds being discussed, though, but the accelerations between (relative to your source/destination) pretty much non-relativistic speeds and the speeds now being discussed. And back again if you actually want to stop at the target. Apart from the fact acceleration produces effects to perceptions of time on top of mere relative velocities as experienced by a coasting frame of reference (unmoving, by your own standards), there's the whole "how do we manage the accelerations needed?" thing.
If there's an "instantly move at light speed" machine, that somehow avoids all the inertial problems (by instantly removing inertia perhaps you'd immediately
go at light-speed..? ...but what related effects are their on your molecular, or even atomic or nucleonic, integrity?) then there's perhaps something useful to think about, but until then there's only
other tricks.
So (@TheBronzePickle), FTL might be done by telling space (which
can apparently distort quicker than SoL) to distorted in such a fashion that you are travelling (locally) at sub-SoL, perhaps even 'slow' enough to not be artificially dilated in time, length or mass at all, but all this within a bubble of altered space-time geometry that is
itself moving through space at FTL speeds. As per the much vaunted Warp Drive idea (from fiction, and since then through somewhat speculative 'fact'). Thus perhaps Picard makes his ship travel... oh, four light years in four minutes (whatever warp speed that would be, probably still somewhere in the 9s, by the tNG measure) and ages four minutes. When someone with a telescope looks at the target system from the origin system four years
and four minutes after Picard left it, he sees Picard's arrival, while a target-system astronomer looking at the origin one four years
less four minutes after Picard's arrival will see his departure. Should Picard hang around for a mere two minutes before setting back to his origin, then he will have aged a further two minutes during that wait and four more minutes during the return trip.
The (original) origin system will see a returning Picard ten minutes after his initial departure, but have still to wait the four years and four minutes from departure (four years
minus those four+two minutes of wait and return from re-arrival), to be able to see the evidence of his arrival, and four years and six minutes from departure (four years minus just four minutes from the return time) to see him start the return journey. Of course, they already know by this time that Picard has
claimed to have done this journey, but now they can get independent visual proof, rather than rely on ships logs and things. (Note that although Picard can say "when I arrived, the sun was just about to go Nova, prepare yourself!", I don't actually see any problem with causality here that the origin planet knowing four years (give or take) from now they're going to be hit by a blast of exploding star, because even if they
could work out how to stop such an event from happening, they couldn't send any mission to the other star any earlier than Picard got there in the first place in order to apply their ready-mix-anti-nova-solution to the star... It's going to happen anyway. Information has travelled faster than light, which is generally taken as a no-no but in a world with Warp Drives is going to be not so absolute, but no information has ever gotten back to an origin
before it left. It always has a positive transit time. (Even instantaneous travel is useless. You only get information, by return of 'insta-Warp' communication, the instant it is sent off.)
Meanwhile, observers at the (mid-point) destination, assuming that their star
doesn't go Nova in the meantime, will as well as having seen evidence of Picard's outward trip starting, four years minus four minutes after his arrival, will, ten minutes later, (those four minutes of transit time, two minutes of wait, four minutes of return-transit time, or just 4y+4m after his return trip starts) see Picard arrive back at his origin again. (If Picard told the mid-point destination that he had just left a gone-nova system, again the scientists at the mid-point destination (advanced enough to have nova-supressing tech, but not to have Warp Drive of their own?) could not get Picard to go back and apply a solution
prior to his having departed with the news. (OTOH, maybe they can give him something, in those two minutes, that can stop the home-nova from getting any worse than it could have gotten ten minutes after his initial mission start with the news that "our sun will be going nova in slightly over ten minutes time... don't suppose you can help us out, eh?"...)
The above assuming that the 'within the bubble' travel is not relativistic enough to add measurable time dilation to the equation Which is probably extreme, but simplifies things (if dilation occurs, then Picard might experience two minutes of travel for four minutes of 'absolute' travel (as measured by respective astronomers tallying arrival and departure times w.r.t. the instantaneous local observations and precisely known 'delayed' observations from afar) across the 4LY of space). It also assumes that the origin and destination are static (for all purposes) with respect to each other.
Start Warping between two mutually moving objects (especially moving fast enough, relative to each other, to show time dilation) and that's when you start wondering when a 'Warped' transport
actually arrives at the other party, and whether or not it could possibly arrive back at you earlier than it left. In my mind, the solution to this is that we obviously cannot use some exotic travelling solution that keeps the traveller's time completely synched with their origin and destination if the origin and destination cannot agree on a single time-reference for which the traveller can remain synched. So if Picard is warping between (say) Sol, and some star like
Gliess 445 on speed, or perhaps
Bellus, that is travelling significantly faster and has a significantly different frame of reference that isn't just 'parallel', then the method by which the transition (either gradual, or at some point between leaving the origin/arriving at the destination) is performed might 'burn up' any "pre-cause effects" time.
This is of course so theoretical (if not heretical) that I'm not saying that it's the way it would happen, or even that it's
possible that this is the way it would happen. For one thing, it relies upon a still rather hypothetical method of travelling, with basic assumptions about how it would be 'experienced' by both travellers and those the travellers depart/arrive at, which might be utterly wrong. Even if the basic idea is sound. As such, I make no apologies for "handwavium" answers or the need fo "unobtainium"-alloyed devices to enact this very premise.