Wikipedia sez that for a body starting at rest at distance $r$, the time it takes to reach a distance $x$ is given by:
$$t(x) = \frac{r^{3/2}}{\sqrt{2 GM}} \left( \arccos(\sqrt{b}) + \sqrt{b(1-b)} \right)$$
where $b=x/r$. The Sun's standard gravitational parameter GM is about 1.327E+20 m^3/s^2.
The time to fall all of the way ($x=0$) is just
$$t(x) = \frac{r^{3/2}}{\sqrt{2 GM}} \frac{\pi}{2} = \sqrt{\frac{r^3\pi^2}{8GM}}.$$
For 1 AU that's about 0.178 years, and for 100 AU it's 100${}^{3/2}$ longer or 178 years.
That means that we'll have to take the motion of the planets into consideration, which is a good thing in this case because we need to hit Earth, not the Sun.
I wrote a short Python script to simulate this. I use a Mickey Mouse solar system with gravitational attraction from the Sun and the four large planets Jupiter through Neptune.
To make the program simple I used "reduced units" with 1 AU = 1.0 and 1 year = $2 \pi$.
In these units:
mass period semi-major axis
Sun 1.0000E+00 - 0.000
Earth - 1.000 x 2π 1.000
Jupiter 9.5476E-04 11.871 x 2π 5.204
Saturn 2.8583E-04 29.666 x 2π 9.583
Uranus 4.3662E-05 84.262 x 2π 19.22
Neptune 5.1522E-04 165.22 x 2π 30.11
I ran the simulation for an initial separation of 100 AU and for many others, with 180 AU also shown below. I can't get beyond 0.003 AU from the Sun's axis, nowhere near 1 AU (Earth's orbit) I dropped a probe every 20 years for 1000 years in order to build up a collection of trajectories.


So you have two choices:
- Start near 1 AU away from the Sun's axis
- Give yourself a small "kick" velocity to one side. At 100 AU it turns out to be about $\pi / 100$ AU/year which is about 150 m/sec, which is a medium-sized delta-v burn.

Python script:
def deriv(X, t):
x, v = X.reshape(2, -1)
positions = np.stack([semis*f(t * omegas) for f in (np.cos, np.sin, np.zeros_like)], axis=1)
xx = x.reshape(-1, 3) - positions
accs = -xx * GMs * (((xx**2).sum(axis=1))[:, None])**-1.5
acc_sun = -x * 1.0 * ((x**2).sum())**-1.5
acc = accs.sum(axis=0) + acc_sun
return np.hstack((v, acc))
import numpy as np
import matplotlib.pyplot as plt
from scipy.integrate import odeint as ODEint
halfpi, pi, twopi = [f*np.pi for f in (0.5, 1, 2)]
GMsun = 1.327E+20
GMs = np.array((1.267E+17, 3.793E+16, 5.794E+15, 6.837E+16)) # m^3/2^2 Sun, Jup, Sat, Ura, Nep
GMs = GMs[:, None] / GMsun
semis = np.array((5.204, 9.583, 19.22, 30.11)) # normalized to Earth's = 1
omegas = semis**-1.5
vs = semis**-0.5
D = 100 # AU
X0 = np.array([0, 0, D] + [0.005, 0, 0.0]) # x velocity is 0.005/2pi AU per year
Tmax = np.sqrt((D**3 * pi**2)/(8))
yearsmax = Tmax/twopi
years = np.linspace(0, 0.999 * yearsmax, 200)
times = twopi * years
answers = []
t_offs = 20. * np.arange(50)
for t_off in t_offs:
answer, info = ODEint(deriv, X0, times+t_off, rtol=1E-11, full_output=True)
answers.append(answer)
if True:
plt.figure()
for answer in answers:
x, y, z = answer.T[:3]
plt.subplot(3, 1, 1)
plt.plot(years, x)
plt.ylabel('x (AU)', fontsize=18)
plt.subplot(3, 1, 2)
plt.plot(years, y)
plt.ylabel('y (AU)', fontsize=18)
plt.subplot(3, 1, 3)
plt.plot(years, z)
plt.ylabel('z (AU)', fontsize=18)
plt.xlabel('Years', fontsize=18)
plt.show()