This may not be a popular opinion, but I can kind of see where Alex is coming from. I know he seems a bit needy, bordering on desperate, but at the same time it doesn't sound like he's been pushing himself on her and forcing her to spend time with him.
Ellie let him think they had a chance then as soon as Alex chose her she's been ignoring him. He's right when he says that's rude.
He does need to calm down a little, step back and give her a chance to talk to him, but I get his frustration; it is Love Island after all. The whole point of it is to find someone. They've all gone in there with expectations of having someone.