The information provided can yield ground speed 200 km/h, and the speed with respect to the ground, but not air speed. Air speed is affected by wind speed, which is not given.
Let V be the actual speed along its climbing path, in earth-frame coordinates. This is probably what you meant by "air speed"
V cos 15 = 200
V = 207 km/h
The climbing rate is
Vy = V sin 15 = 53.6 km/h.
To climb 1000 m (1 km), the time required is t = (1 km)/Vy = 0.0187 h = 1.2 minutes
"An airplane is climbing at an angle of 15 degrees to the horizontal with the sun directly overhead. The shadow of the airplane is observed to be moving across the ground at 200 km/h. How long does it take for the plane to increase its altitude by 1000 m"?
There was another part to this question that I had figured out. I had calculated the actual airspeed of the plane, which was 207 km/h.
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