Riemannian geometry is a large field which still has active research going on, and much of it is applicable to PDE. I doubt there's a good answer to this question, because it will depend very closely on what you want to do. Different areas of PDE will use different techniques, and unless you have a very good idea what you want to do there's no way to know what you will need. If you have an advisor, they would be the best person to talk to regarding this.
To quantify a little bit, at risk of loss of generality, you probably wouldn't get much out of a month of study if you know nothing now. At my school we have a 2 semester graduate class in differential geometry, which is essentially the minimum you'll need if you want to do anything serious. Riemannian geometry was about 70% of the curriculum. We also have a 2-semester graduate class in PDE, which again I'd guess is the minimum one needs to do anything serious there. There's no reason you have to take a class, but one way or another you'll need to learn it, so the amount of time should be approximately the same.