Wow! What an opportunity!
I think a mountain bike, or a hybrid, would be in order. While they race bicycles on cobblestones in Europe, I wouldn't think it would be a tolerable measurement on a road bike, going across all the Great Wall bricks.
He didn't say how long of a course, so I don't know now many staircases he may encounter. There was a discussion a while ago regarding measuring up and down steps. I believe, if the staircase is not too long (subjective), you can roll the bike up and down. If it is too long, a tape from top to bottom to give a straight-line length, then ride the bike the appropriate distance before or after the staircase.
Some of this depends on the length of the course. A 5k with one staircase may be approached differently than a Marathon with 50 staircases.
Of more concern, at least from where I sit, is all the tourists to dodge.