The 12th Planet's path is elliptical, making a long flat circle around its two gravitational masters, your Sun and a body you cannot see. The Sun's alter ego in this matter is not an object on your sky maps, but for the purposes of calculating the 12th Planet's orbit, you can assume it be have the same mass as the Sun, and to be at a distance that allows the curve of the ellipse to smooth to an essentially straight line between the two orbital foci. The 12th Planet's travels are not unlike a train on parallel tracks, where the train is on one side of the tracks going in one direction, and on the other side coming back. It will surprise you to know that the second foci is not that far away. Since it rivals the Sun in mass, the assumption would be that your astronomers would know about it. However, being dark, they stare past it and think it space. To use multiples of the distance from your Sun to its farthest known orbiting planet, which you call Pluto, this foci is from the Sun 18.724 times as far away.
The elliptical path of the 12th Planet does little direct damage to the planets in your Solar System, which are lined up on a plane with each other. The 12th Planet comes in on an angle, such that it is only the Point of Passage where a direct collision could occur. This point, at the present time, is not in the orbital path of any of your Solar System planets, although that was not the case in the distant past as your Asteroid Belt attests. For this pole shift, we estimate that the Earth will be on the same side as the 12th Planet, at approximately an 18 degree angle from where the 12th Planet comes closest to the Sun. Its return is on the other side of the Sun, but for the Earth, hapless in this matter, this may be an advantage or disadvantage depending on where the Earth is in its own orbit at the time. On the 12th Planet's return voyage, during this passage, the Earth will be protected by the Sun from a full impact, but mild earthquakes and heavy tides will recur at that time.
Having passed by the Sun, the 12th Planet now slows. The rate of slowing is dependent on two factors, essentially - its speed and the fact that both its gravitational masters are now behind it. As fast as the 12th Planet picked up speed approaching your Sun, it slows even faster, the nearness of your Sun behind it no small factor in this. Nevertheless, for a traveling planet the size of the 12th Planet, putting on the brakes and turning about is no small matter. It must first come to a stop, which it does in approximately 2 years 3 months after passing your Sun. The 12th Planet's orbit takes it well away from the Sun after passage, so that it moves out a distance equal to 1/4 of the distance between the Sun and its other foci before it slows to a stop. After passing through the Solar System, the 12th Planet moves out on the opposite side some 3.560 times the distance from your Sun to its farthest planet, Pluto, then stops. It then hovers, not moving, essentially, for 3 years 6 months, and then slowly begins a return trip which telescopes or mirrors the voyage out.
There are no stars in the sighting line that can be used as a guide, but one need not grope about in the heavens for guide posts when the best are within your own Solar System. Neither the 12th Planet's orbital plane or any axis placed upon it at intersection points have identifiable stars directly on the plane or such an axis. It would be a bit to this side or a bit to that side, and there we are again with imprecision. Let us suggest that a more satisfying approach would be to use stabilizing points within your Solar System. The path of your Moon is well known and its orbit forms a plane as it moves around your Earth. Even though your Moon circles the Earth and the Earth circles the Sun, the orbital planes are at an angle to one another and thus the Moon's orbital plane consistently intersects the Earth's orbital plane at opposing points twice a year. This will be stable until the cataclysms are upon you and thus can be used to triangulate with other planes.
The Earth's orbit forms a plane. The Moon's orbit forms a plane that bisects the Earth's orbit in a fixed place twice a year. The 12th Planet's orbit, coming and going, forms a plane that also bisects the Earth's orbital plane. The 12th Planet's orbital plane can be calculated if points are taken on the other two planes and used as a reference. The Earth's distance from the Sun is known. Take the placement of the Earth at the two points where the Moon's orbital plane lines up. Use these two points as two of three points in a triangle. The third point in an equilateral triangle will be on the plane of the 12th Planet's orbit. This third point is more stable than any point we could give you where you would be looking out into space. Recall that the 12th Planet is lifting up and away from the Sun when it is this close, so will not actually be at this third point as it approaches. However, for purposes of calculating the orbit in the heavens, this third point should be useful.