TL;DR: 47 minutes and 20 seconds at constant one-gee felt by the passengers.
An important thing to consider in this sort of problem is that there are, in the rotating reference frame, five separate forces we need to consider: gravity, centrifugal force, Coriolis force, drive, and the constraining force of the cable.
Given as constant the mass of Mars and its sidereal period, and assuming a spherical Mars, this gives us the forces varying as follows:
Gravity varies inversely with the square of the distance from the center of Mars ("radius") and acts downward.
Centrifugal force varies directly with the radius and acts upward.
Coriolis force varies directly with the speed of the car and acts laterally.
Drive is controllable and acts upward or downward as needed.
Constraining force is exactly counter the Coriolis force and is the reaction of the cable against the Coriolis force produced by the car's velocity.
Now let us assume that the passengers will prefer to feel one-gee pressing them to the floor the entire trip. This does not mean that they are accelerating at that rate. Just consider vertical forces momentarily. Gravity acts against that one-gee, weakening with altitude, and centrifugal force acts with the drive force to pull the car upward. 461
Once the car begins to decelerate, gravity helps despite its weakening strength, and centrifugal force pulls against the floor of the car. Now by definition, areostationary orbit (the point we are trying to reach) is at the altitude where gravity and centrifugal force exactly counter each other, so we will see the total surface-ward force on the car before applying the drive vary from Martian surface gravity down to zero at our resting point.
Taking into account only those vertical forces brings us very close to the 47-minute figure already mentioned. However, by numerically integrating (one second intervals on a trapezoidal integration) and examining at various points I found that the velocity of the car produces a Coriolis force at maximum speed of around 12 km/s comparable to Martian surface gravity. That's a significant force pulling the car away from the cable, and the constraining force is felt by the passengers, with the vector sum of that with the drive force at higher than one-gee, angled away from the floor slightly. So instead of a constant one-gee drive, why not have a car that can angle itself (it would have to anyways to flip and decelerate), and have the car's drive vary as needed so that the total force on the passengers is one-gee?
By integrating all these forces over time, and using the simplification that rotating the car takes only one second, we find that at 26:00 the car reaches maximum speed of approximately 12.5 km/s, with a lateral force component of 1.78 m/s^2 (0.18 g), flips and begins decelerating, then comes to a stop at the areostationary orbital point of the elevator at 47:20.
As to the issue of a tram being able to travel that fast, and it being considered safe - this is in essence a space ship constrained to travel in two directions, and 12.5 km/s is comparable to other velocities of space travel. 461