Does the pilot of a rocket ship experience an asymptotic approach to the speed of light?

The equations you need are given in John Baez's article on the relativistic rocket. If $T$ is the time measured on the rocket, and $V$ is the velocity measured on the rocket, then the equation you need is:

$$ V = c \tanh \left( \frac{aT}{c} \right ) $$

For a rocket accelerating at $g$ the velocity time looks like:

So, as you guessed, option (1) is the correct answer.

Bear in mind that the crew of the rocket assume they are stationary and that the rest of the universe is accelerating towards them. Their situation is as if they were stationary in a constant gravitational field of $g$ and were observing everything else falling towards them in that field. It should come as no surprise that they do not observe any superluminal velocities.

If you're interested in taking this further, the coordinate system for a uniformly accelerating observer is called the Rindler coordinates.

Incidentally, one of the few things observers in two frames will agree on is their relative speed (measured as the two observers meet i.e. when they're at the same point in space). You'll find that the time dilation and length contraction balance each other out to give both frames the same relative speed.