I will attempt to answer your questions but I have not heard some of them before.
1. he is an exponent of spin bowling. ??? Could this be He is an expert in spin bowling. Spin bowling is a cricket term I believe. and you can find an explanation here: Spin bowling - Wikipedia, the free encyclopedia
2. back then means historically or during a time before the present.
3. "the eye-wateringly expensive prices" are prices that are so high that they make you want to cry.
4. "come good on your promises" you might also hear as: make good on your promise and it simply means that you should fulfill your promises.5. "competitive edge" is an advantage in business or a competition
6. "he cut a sorry figure in the examination" I have not heard this one before but I imagine the meaning is that he did not do well in the examination.
I hope this helps,
Student or Learner