Null geodesics are of interest because their collection can give you the set of null-cones on the spacetime, which, in turn show the constraints on massive particle worldlines. I think what John Rennie was saying in the comments is that once you see the alignment of null cones in Kruskal-Szekeres coordinates, it is clear that a time-like worldline, after crossing the event horizon, will end at the singularity. And, because there is no compactification for KS coordinates (like on a Penrose diagram), "it is obvious" that this end must occur in finite proper time.
To calculate the infall time you need to integrate the geodesic equation. You could, e.g., follow these lecture notes by Joel Franklin (Reed College) on radial freefall. They show that the proper time in going from some initial coordinate $r_0$ to some final value $r$ is:
$$
\tau(r) =\frac{ \pm 2}{3 \sqrt{2 M}} \left( r^{3/2} - r_0^{3/2} \right)
$$
where the sign is chosen to be "$-$" for infall (and the initial velocity, $dr/d\tau$, has been chosen specially to get this simple form; see comments of Cham below). Thus the infall time to $r=0$ is
$$
\tau_\text{infall} = \frac{2 \, r_0^{3/2}}{3 \sqrt{2M}} \quad \rightarrow \quad c \, \tau_\text{infall} = \frac{2 r_0^{3/2}}{3 \sqrt{ 2 M \left( \frac{G}{c^2}\right)}} \quad \rightarrow \quad \tau_\text{infall} = \frac{2 r_0^{3/2}}{3 \sqrt{2 G M}}
$$
where I've followed Wald's prescription for returning to physical units. Alternatively, you could write this as:
$$
c \, \tau_\text{infall} = \frac{2}{3} \sqrt{\frac{r_0^3}{r_s}}
$$
where the Schwarzschild radius is $r_s = \frac{2 GM}{c^2}$. Taking the initial point to be the Schwarzschild radius, $r_0 = r_s$, the time inside the black hole is:
$$
\tau_\text{inside BH} = \frac{2}{3} \frac{r_s}{c} = \left\{\begin{array}{ll}
\frac{2}{3} \frac{2950\,\text{m}}{c} \approx 6.6\, \mu\text{s} & M = 1\, M_\odot\\
\frac{2}{3} \frac{1.2\times10^{10}\,\text{m}}{c} \approx 40\, \text{s} & M = 4\times10^6\, M_\odot\\
\end{array} \right.
$$
Figure 31.1 in those lecture notes nicely shows the finite (proper) time of infall for the object, as compared to the infinite infall time as seen by an observer at infinity.
And there is no need for the infalling worldline to traverse any "distance". Once you cross the event horizon, the future singularity will arrive to you in that finite proper time. You don't need to go anywhere to find it.
