There was no guarantee that he wasn't going to puke, but he was going to try really hard not to. He was even more worried about it because Shuri asked him not to.
The entire trip up into space, Ned did not have his eyes open. His eyes were shut tight and he murmured the mantra of 'Do NOT throw up' over and over to himself. Hopefully Quill wouldn't need anything pulled or done from him because Ned was basically trying to stay cool.