The term gentleman jockey can be used as a derogatory term for a member of the upper class dabbling in the sport - a dilettante with no real skills.
An excellent description can be found in the January 1848 issue of The Sportsman:
Critics will tell us that at the amateur performance, Mr. So-and-so,
or Lord Such-a-one, played Charles Surface or Sir George Airy,
very well--for a gentleman. Even in our own line, where a gentleman is
declared to be nothing and nobody without his recreations, he still
plays but second ﬁddle at them. A gentleman huntsman, nine times in
ten, is taken but as another term for a bad one; and a gentleman
jockey generally unites the two on much the same understanding. At
most he may reach in this sphere the acme allowed by Mr. Scrope
Davies, who, in speaking of a friend’s great efforts to carry out the
character in every perticular, admitted “he did look and ride like a
jockey, but then it was like a bad jockey."
The Sportsman, January 1848, p. 343
I could not find any occurrences of the phrase gentleman jockey wins the Derby other than the dialog mentioned in the OP. But based on the above I would interpret it as an affirmation and rephrase of the preceding sentence:
01:14:56 I'm just an amateur.
01:14:59 Beginner's luck, huh?
01:15:02 Gentleman jockey wins the Derby.
01:15:05 Something like that.
In other words, my success is due to an unlikely stroke of luck, similar to an amateur (an inept one) winning a professional competition.