You can get a first approximation to the Clopper-Pearson interval by assuming the distribution is symmetric, in which case the events Y <= 2 are "just as extreme" as the events Y >= 15-2 = 13, and your approximate p-value is
p-value_1 = Pr(Y <= 2) + Pr( Y >= 13)
A glance at your picture reveals that symmetry wasn't a good assumption, as the event Y = 13 is quite a bit more likely than the event Y = 2. Indeed the event Y=14 is also more likely than the event Y=2. But the event Y=15 seems (from your picture) to be less likely than the event Y=2, so you might include it in your second-approximation to the p-value calculation (which should sum the probability of your observed event and of all "more extreme" events):
p-value_2 = Pr(Y <= 2) + Pr( Y >= 15)
For your third (and final) approximation, you'd check the probability density at Y=15. I'm confident a_statistician got it right, with Pr(Y=15) < Pr(Y=2), so your two-tailed p-value is
p-value = Pr(Y <= 2) + Pr( Y >= 16)
= Pr(Y <= 2)
As pointed out in Two-sided binomial test in Excel, the Clopper-Pearson 2-sided binomial test isn't something you'd want to perform in Excel. You can hack your way through it for particular cases such as the one in your diagram. Someday, someone somewhere will go to the trouble of developing and carefully-validating and publishing a Visual Basic routine BINOM.TEST.TWOSIDED. But... why not learn S, or pay the licensing fees for SPSS?