An equivalent condition for an element to be integral
Some ideas that should help.
$\Rightarrow$. If $u$ is integral over $R$, then $u$ satisfies some monic polynomial, say of degree $n$, with coefficients in $R$. Show that any nonnegative power of $u$ can be written as an $R$-linear combination of $1, u, \ldots, u^{n-1}$. It follows that you really just need to find an $r$ such that $ru^i \in R$ for $i < n$.
$\Leftarrow$. The hint is good. $R\frac1r$ is certainly finitely generated as an $R$-module, so it's Noetherian. Show that $R[u] \subset R\frac1r$ using the given condition. What do you now know about $R[u]$ as an $R$-module?
The nontrivial direction is a special case of the Lemma below, with $ \,D =\,$ integral closure of $ \,R.\,$
Lemma $\ $ Suppose that $ \,D = \mathbb Z\,$ (or any Noetherian integrally closed domain, e.g. any PID), and suppose that $ \,w\,$ is a fraction over $ \,D\,$ such that some unbounded sequence of powers of $ \,w\,$ has a common denominator $ \,0 \ne d\in D,\,$ i.e. $ \,d\!\,w^{n_i}\in D\,$ for all $ \,n_i.\,$ Then $ \,w\in D.$
Proof $\ $ By ACC the sequence of ideals $ (d, dw^{n_1}, dw^{n_2},\ldots)$ eventually stabilizes, which implies that for some $ \,k\,$ we have $ \, dw^{n_k}\in (dw^{n_{k-1}},\ldots, dw^{n_1}, d),\,$ which implies
$$ d\, w^{n_k} + c_{n_{k-1}} d\, w^{n_{k-1}} +\,\! \cdots +\, c_{n_1} d\, w^{n_1} + c\:\! d\, =\, 0$$
Cancelling $ \,d\,$ yields $ \,w\,$ is integral over $ \,D,\,$ hence $ \,w\in D,\,$ since $ \,D\,$ is integrally closed. $\ $ QED
Remark $\ $ Fractions whose powers have such a common denominator are called almost integral. It is clear that integral fractions are almost integral, since if $\,w=a/b\,$ is a root of a monic polynomial of degree $n$ over $D$ then all powers of $w$ lie in $\, D[w] = D\langle 1,w,\ldots,w^{n-1}\rangle$ so $\,b^{n-1}\,$ is a common denominator for all elements of $\,D[w].\,$ The above proof shows that the converse holds true in Noetherian domains, i.e. $ $ almost integral $\iff$ integral, in Noetherian domains. This is employed implicitly in Dedekind's pioneering work on ideal theory.
Here is a typical application from my answer to this prior question.
Show $ \ a\mid b^2,\, b^3\mid a^4,\, a^5\mid b^6,\, b^7\mid a^8 \,\cdots\,\Rightarrow\, a = b\,$ for $ \,a,b\in\mathbb Z_+$
Hint $ \ \ \forall\, n\in\mathbb N:\ \ a\,\!\left(\dfrac{a}b\right)^{4n+3}\!\in\mathbb Z,\,\ b\,\!\left(\dfrac{b}a\right)^{4n+1}\!\in \mathbb Z\ \ \Rightarrow\ \dfrac{a}b,\,\dfrac{b}a\in\mathbb Z\ \ \Rightarrow\ \ a = \pm b\ \ \ $ QED