I understand that Plato thinks our knowledge of universals is innate, and therefore synthetic a priori knowledge. However, does he think we can have any knowledge that is synthetic and a posteriori? For example, is my knowledge that "the weather is sunny today" also in my mind since birth (and is just "revealed" in me through reason), or does Plato accept that some synthetic knowledge can be acquired through sense experience?
It's a good question (philosophers often retroactively apply terminology), but not suitable for this site.
p28 of 'Handbook of Epistemology' says that Plato (his "episteme"), like Leibniz,
admits only a priori truths, i.e. only a analytic and synthetic a prori as legitimate pieces of knowledge.
If you scroll up a paragraph, it says that Leibniz believes in
truths concerning facts [that] are synthetic a posteriori. However, this is so only from a human epistemic perspective, since for God all truths are analytic, and therefor a priori.
So according to the authors I. Niiniluoto, Matti Sintonen, Jan Wolenski, Plato does not believe in synthetic a posteriori knowledge, but that may not mean that what we take to be synthetic a posteriori knowledge is not knowledge.
I have not read this book, and am merely suggesting you can google it to see that the claim he does not believe in them is not trivially wrong, as I have. If you want a more thorough answer you could start with that and look into the difference between episteme (knowledge) and techne (craft), as I would have guessed/thought he believed in a posteriori art (I suggest googling that again to exclude the same triviality to how you are confused). Though again, google tells me that for Griswold, Plato's "techne" is a retrospective and
a posteriori way of organising something one already knows... calling into question the tyranny of the Platonic sign that is always and already transcendent and a priori
No-one here is going to be able to give a definitive answer, but here is I hope a coherent one.