To suck it up means to endure something painful or unpleasant, and in this context the speaker is considering enduring the latter in order to hear what the other person has to say.
Consider the staetement "I'm stuck between the desire to run away quickly to memorial church, the nearest hallowed ground I can think of-or to endure and hear what he's going to say this time.". Based on this statement, is "I'm torn between the desire to run-hightail it straight to memorial church, the nearest hallowed ground I can think of-or to suck it up and hear what he's going to say this time." a true statement? Yes/no/unknown