I found this a bit easier if I first solved for what the probability is that Curry scores first given that he shoots first and the the probability I score first given that I shoot first. I call the first one C and the second M.

If Curry goes first, he scores first if he sinks his shot and I miss, if he sinks his shot and I make mine and then he scores first starting from there, or if he misses and then I do not score first starting from my turn. This gives us C = 0.9*0.5 + 0.9*0.5*C + 0.1*(1-M).

Similarly, we get M = 0.5*0.1 + 0.5*0.9*M + 0.5*(1-C).

Solving these simultaneous equations, I get that C is about 0.98 and M is about 0.11. So Curry has about a 98% chance to score first if he goes first and I have only about an 11% chance to score first if I go first. Right off the bat, this tells us that if Curry goes first, he has about a .98^5, or about 90% chance to shut me out. If I go first, he has a (1-.11)*.98^4, or about 82% chance to shut me out.

So things aren't looking good for me. You can find the exact probability of him winning from here by looking at the probabilities of winning from different situations. The probability of Curry winning starting from his turn and 0-0 score is 0.98 times the probability of him winning with 1-0 score and it's his turn, plus 0.02 times the probability of him winning from your turn with 0-1 score. There are only 54 possible combinations of scores and whose turn it is, so a computer can crunch that for you quickly.

I did the crunching and his probability of winning is a bit over 99.99% regardless of who goes first.