You need to sit down your roommate and come to an agreements about this that works for the both of you. This means you need to talk about it together and more importantly learn to resolve situations straight on if you want this roommate stuff work smoothly for the both go you and that's how you need to sit her down.
You are obviously different people, with different values and you should be able to come to a compromise where both can feel good with. This requires both of you to give in at times and an attempt to understand each other, but more importantly respect each other and thus talk things through.
If you start the conversation something like this and ask her to be open to your side of things as well, you're likely to get her to work with you and you two indeed can come to some sort of resolution. You are not a neat freak nor should you be forced to live up to her expectations. You are right, rinsing the bowl can be enough, although if you're using this water you might as well clean it completely (only 2 more actions, soap and towel), but that's beside the point here. If she wants the bowl to be clean or thinks that rinsing isn't enough then she will have to clean it up herself. It's as simple as that.
That doesn't means she doesn't get anything at all, because you can attempt to be a little better at your time management so you indeed can do the whole tast. As I pointed out, it's not that much extra work to do it all and you can agree to try so she feels good too. However you, like her, need to get the space to let things slip or slide on occasion if indeed you're late or something else comes up. It happens from time to time and you should not have to endure a meltdown because of one shitty bowl. Especially not if she can take care of the problem as easily herself. And of course the same applies to you too, not giving her a meltdown over something if you two should talk about.
This is the start of you two living together. To make it liveable together you need to make agreements together. The faster you do this, the better and easier the rest of the year and possibly longer will go between the two of you.
It's good practice in standing up for yourself, and what's right too, learning to negotiate (something you'll need later on) and it can help improve your communication skills, which is another valuable skill. So sit her down and have a talk about this.
If she is unwilling to work with you, ask for someone who's in charge of housing to mediate. And if that doesn't work you can always see if you can't get another roommate. But try to work things out together first. Talk to her and then see what your next step is, because no you should not have to walk on eggshells just because your new roommate is such a neat freak. It's your home too and you should be able to feel at home too.