It is not true that the right ring of fractions always exists in the setting you describe. An example where it does not can be found in section 2.1.7 of McConnell and Robson's book Noncommutative Noetherian Rings.

They take the enveloping algebra $U(\mathfrak{g})$ of the $2$-dimensional $k$-Lie algebra $\mathfrak{g}=kx\oplus ky$ with $[xy]=y$ (here $k$ is some field) and show that the set of elements $S$ of $U(\mathfrak{g})$ not in the 'augmentation ideal', ie the kernel of the natural counit map $U(\mathfrak{g})\to k$, is multiplicative but does not satisfy the right Ore condition. More precisely they show that $yS\cap (x-1)R=\emptyset$.

Since every non-zero element of $U(\mathfrak{g})$ is regular and $U(\mathfrak{g})$ is both left and right Noetherian this suffices.

This example (along with many others) is developed at greater length in Jategaonkar's monograph Localisation in Noetherian Rings (see 1.3.17 for this one). This should be the starting point of any serious study of these kind of questions.