Solution:

If he runs up 8 steps, then he needs 22.5 seconds to reach the top.
If he runs up 14 steps, then he needs 13.5 seconds to reach the top.

The 6 additional steps take 9.0 seconds.
Therefore, each step takes 1.5 seconds.

Total steps in escalator = 8 + 22.5 / 1.5 = 23 or Total steps in escalator = 14 + 13.5 / 1.5 = 23.

If Brian did not run up any steps at all,
he would reach the top of the escalator in 34.5 seconds (i.e., 23 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 − 8)/22.5 = (x − 14)/13.5

The above equation may be solved as follows.

13.5 (x − 8) = 22.5 (x − 14); or
x = (14 × 22.5 − 8 × 13.5) / (22.5 − 13.5) = 23.

Now, Speed of escalator = (23 − 8)/22.5 = (23 − 14)/13.5 = 1/1.5 steps/second.
Time to reach top = Total Steps / Speed = 34.5 seconds.

Food for thought:

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