This answer is about the last part on how to extend the duality beyond $\sigma$-finite measures since the first part is taken care of in If $f$ is measurable and $fg$ is in $L^1$ for all $g \in L^q$, must $f \in L^p$? mentioned by David Mitra.

The answer to "what is the best way to prove that $L^{q}(\mu)$ is the dual space of $L^{p}(\mu)$ even if $\mu$ is not $\sigma$-finite" is probably a matter of taste. I think the proof using uniform convexity and the Milman-Pettis theorem suggested by David Mitra is nice. The usual expositions apply Clarkson's inequalities or Hanner's inequalities. The latter are easier to prove than the former, but they leave me with the feeling of coming out of the blue, see also the discussion on Mathoverflow.

An illuminating discussion of the approach to duality via uniform convexity of the $L^p$-spaces is given by Harald Hanche-Olsen, On the uniform convexity of $L^p$, Proc. Amer. Math. Soc. 134 (2006), 2359-2362. The paper ends with an outline of the proof you're asking about. Alternatively, you can consult the references provided by David Mitra in the comments.

However, I doubt that the above-mentioned proofs are what Rudin had in mind in this exercise. After all, we want to prove that $L^q$ is dual to $L^p$ for $\frac{1}{p} + \frac{1}{q} = 1$ by using the methods he provides in said chapter and by reducing to the $\sigma$-finite case.

It is not hard to prove that the natural map $L^q \to (L^p)^\ast$ given by sending $g \in L^q$ to $\varphi_g(f) = \int f g$ is isometric: this uses Hölder's inequality and one verifies that $f = \lVert g\rVert_{p}^{1-p} \frac{\lvert g\rvert}{g}$ is an $L^p$ function of norm $\lVert f \rVert_q = 1$ such that $\varphi_g(f) = \lVert g\rVert_q$.

It remains to prove that every $\varphi \in (L^p)^\ast$ is of the form $\varphi_g$ for some $g \in L^q$.

Observe that a function $g \in L^q$ must vanish outside of a set of $\sigma$-finite measure (since $\lvert g\rvert^q$ is integrable). So, given a continuous functional $\varphi \colon L^p \to \mathbb{R}$, we must somehow be able to prove that it is "supported" on a set of $\sigma$-finite measure. By this I mean that we must be able to find a $\sigma$-finite set $E$ such that $f|_E = 0$ implies $\varphi(f) = 0$. If we have that, the known duality applies.

Let $f_n \in L^p$ be functions of norm $\lVert f_n\rVert_p = 1$ such that $\varphi(f_n) \to \lVert \varphi \rVert$. The intuition is that $\lVert \varphi\rVert-\varepsilon \lt \varphi(f_n) \leq \lVert \varphi\rVert$ means that $\varphi$ must almost be zero on functions whose support is disjoint from the one of $f_n$.

Collect all the supports of the $f_n$'s together into one set $E$, using a small trick to show that it is $\sigma$-finite: let $E_n = \{x \in X : \sum_{i=1}^n |f_i(x)|^p \geq 2^{-n}\}$. We have $\mu(E_n) \lt \infty$ and $E = \bigcup_{n=1}^\infty E_n$ is a set of $\sigma$-finite measure containing the supports of all the $f_n$'s. If we manage to prove that $\varphi(h) = 0$ for all $h \in L^p$ such that $h|_E = 0$ we're done: $E$ is $\sigma$-finite and thus we find a function $g$ in $L^q(E)$ representing $\varphi$ (extend $g$ by zero on $X \setminus E$).

Notice that $L^p(X) = L^p(E) \mathbin{\oplus_p} L^p(X \setminus E)$, by which I mean that each $f \in L^p(X)$ can be uniquely written as $f = f|_E + f|_{X \setminus E}$ and we have the identity
$$\lVert f \rVert_{p} = \left(\lVert f|_E \rVert_p^{p} + \lVert f|_{X \setminus E}\rVert_{p}^p\right)^{1/p}$$
for the norm. By duality and the assumption $1 \lt p \lt \infty$ we get
$$\lVert \varphi\rVert = \left(\lVert\varphi|_{L^p(E)}\rVert^{q} + \lVert \varphi|_{L^{q}(X\setminus E)}\rVert^q\right)^{1/q},$$
see $(X \oplus_p Y)^*$ isometric to $(X^*\oplus_q Y^*)$ for details. By the choice of $E$ we have $\lVert\varphi\rVert = \lVert \varphi|_{L^p(E)}\rVert$ whence $\varphi|_{L^p(X \setminus E)} = 0$, as claimed earlier.

To repeat, $E$ is a $\sigma$-finite set, we can find $g \in L^q(E)$ such that $\varphi_g = \varphi|_{L^p(E)}$ and by setting $g = 0$ on $X \setminus E$ we get a function in $L^q(X)$ representing $\varphi$. Thus, the map $L^q(\mu) \to L^p(\mu)^\ast$ is surjective.