Of course the sun can rise directly ahead of the aircraft. It means the pilot isn't necessarily flying at 90 degrees or 270 degrees, but a bit less.
So the plane starts out flying East (well, slightly south of due East), always keeps the sun ahead and ends up flying West in the end.
At 360 knots, the distance travelled during a day (if the 15.5 hours or so quoted by Mjzair are correct) will be just above 10,000 kilometres. That may well put the plane in a place where the sun sets at a different time, especially as it will spend most of the day flying southwards (with some east- or west- element in it). If the plane travels a quarter of the earth's cirumference in distance, and a lot of that southwards, it could end up south of the Equator. On the other hand, because it starts out flying East, it is speeding against the sun, so the day will be shortened (and in the evening, when it chases the sun, the day will be lengthened. If there is no wind - and you have to assume there is none - then the two should be exactly equal in effect and the day ends up at the same length. That would put the plane back on the exact same longitude as it started out as, just much further South (unless it actually crossed the Equator, and flew north for a while)) So if this is correct (and it may not be) you know at least one thing about the flight path: Start and End will be along the same longitude.
Now what happens in between? My initial guess was a sort of semi-ellipse, but I suspect I'm wrong, because the thing will be distorted by the location of the sun.
Hmmm. Let me think. Let's start with three lines: a vertical one (for the start & end location) and the 23.5th parallel (the path of the sun) and the 45th parallel (for the plane). Add two lines: One, 20,000 km to the East, and one 20,000 km to the West. Where these intersect the 45th parallel, the sun will rise and set. The sun travels at a constant speed. Maybe the easiest way to solve this would be with a computer program? Numerically? I know I'm ignoring the altitude of the plane and the Earth's curvature, but I think for a first estimate, this should be a feasible approach. Program it so that there's two points, at two coordinates. Let one travel on a constant path at constant speed, and let the other always head towards the first, at constant speed but naturally changing direction at every time step. Set the time step to ten minutes, and the total time to 12 hours. (err, why is the damn day longer than 12 hours? I hate 3D!!!!).
Ah, let's just say I failed miserably...
(Now wait for people more clever than myself to come in and say how wrong I am!