Taking into account that we know that gravity and acceleration are fundamentally indistinguishable from each other, Would an observer in a black hole believe that he's accelerating upwards faster than light as an explanation as to why light cannot escape? Would he see that it is stationary? Do any of these things violate General relativity?
The physics inside the event horizon of a black hole are... interesting. To answer your question: no, nothing travels faster than light, ever. Instead, we have to re-negotiate what time means as the relativistic effects increasingly dominate the intuitive classical mechanics. As for what relativity says will happen, consider an excerpt from this paper:
Far from the event horizon, the coordinate t approximates an observer's proper time or wristwatch time. This leads us to think of the coordinate t as representing time. This is not true at the event horizon, however. As an infalling observer nears the event horizon, the coordinate t has less and less to do with time as he perceives it – that is, his proper time. In order to understand how time is perceived by the infalling observer, we need to focus on his proper time and ignore the coordinate t.
Even though we can never actually see someone fall through the event horizon (due to the infinite redshift), he really does. As the free-falling observer passes through the event horizon, any inward-directed photons emitted by him continue inward toward the center of the black hole. Any outward-directed photons emitted by him at that instant are forever frozen at the event horizon. So the far-away observer cannot detect any of these photons, whether directed inward or outward.
The paper also explores a slightly fanciful concept of trying to describe what it might look like inside the black hole, without violating any of the math which they did earlier in the paper:
Life Inside the Black Hole
Some have speculated that our universe might exist inside a gigantic black hole. Let's explore this idea further, in order to gain more insight into what the interior of a black hole is really like (and for the fun of it). If our universe is inside a giant black hole, one might ask where the event horizon is. Is there any path we can take that will bring us closer to the event horizon? According to general relativity, if our universe is inside a black hole, every point in our universe is moving closer to the center of this black hole, and away from the event horizon. There is no (spatial) direction that will bring us closer to the event horizon. As it is difficult to visualize a four-dimensional curved surface, subtracting a dimension or two makes it easier. Imagine a giant sphere, and a point on the interior surface of this sphere. This point detaches from the inner surface and moves toward the center, at the same time expanding into a disk. This expanding disk represents our universe expanding in space as it moves through time. In this model let's suppose that our universe formed on the event horizon of the giant black hole, represented by the surface of the sphere. We suppose that the Big Bang occurred at the event horizon of the black hole. See figure 6. The expanding disk is a two-dimensional representation of the three spatial dimensions of our universe. (We could label these spatial dimensions θ, φ, and t.) Every point in our universe (the disk) is moving away from the inner surface of the sphere (the event horizon) toward the center of the sphere (the singularity of the giant black hole). The dimension through which this disk is moving is a timelike dimension (which we could label r). For every point on the disk (our universe at a point in time), the event horizon lies in the past and the singularity of the black hole lies unseen in the future. All timelike and lightlike world lines in our universe lead from the event horizon to the singularity of the black hole. To travel to the event horizon would be to travel backward in time. Therefore, there is no path we can take that will bring us closer to the event horizon