Solution:

If he runs up 7 steps, then he needs 46.5 seconds to reach the top.
If he runs up 14 steps, then he needs 36.0 seconds to reach the top.

The 7 additional steps take 10.5 seconds.
Therefore, each step takes 1.5 seconds.

Total steps in escalator = 7 + 46.5 / 1.5 = 38 or Total steps in escalator = 14 + 36.0 / 1.5 = 38.

If Robert did not run up any steps at all,
he would reach the top of the escalator in 57.0 seconds (i.e., 38 steps × 1.5 seconds/step).

Alternative Solution through Equations:

Let the total number of steps in the escalator be x.
The escalator moves at a constant speed given by

Speed of escalator = (x − 7)/46.5 = (x − 14)/36.0

The above equation may be solved as follows.

36.0 (x − 7) = 46.5 (x − 14); or
x = (14 × 46.5 − 7 × 36.0) / (46.5 − 36.0) = 38.

Now, Speed of escalator = (38 − 7)/46.5 = (38 − 14)/36.0 = 1/1.5 steps/second.
Time to reach top = Total Steps / Speed = 57.0 seconds.

Food for thought:

Does it matter whether the escalator is moving up or down?