I assume there is only Earth and the ship in your system.
For one thing the ship will not fall into orbit just like that. While
falling, it gathers kinetic energy which will allow it to leave Earth
again towards outer space, after its trajectory has been deflected by
Earth gravity. The amount of deflection depends on its speed and how
close it comes to Earth. The shape of the trajectory is that of (half)
(source: Wikimedia Commons)
Assuming the ship has a long axis (i.e. it is not a homogenous
sphere, or concentric homogenous spherical shells), it will be subjected to a rotation because of tidal forces.
Tidal forces are simple to understand. The ship is
supposed to be a reasonably rigid and strong solid (no need for a General Products
hull though). The trajectory of the center of mass of the ship is the same as if the
whole ship mass were concentrated there.
However, that is not the case. Some parts of the ship are closer to
Earth than the center of mass, while others are further away. So the
parts that are closer to earth tend to be deflected more than the parts
that are further from earth. Or to see it another way, they will be
more attracted by Earth. This creates a torque that makes the ship
rotate around its center of mass, and tries to align its long axis
with he center of gravitational attraction.
How fast this rotation takes place is dependent on actual figures.
Note that the ship follows a curved hyperbolic trajectory. So, even
assuming that, at some point, the long axis is right on (the tangent
to) the trajectory, thus balancing the gravitational effects between
both ends of the ship, this will no longer be the case a bit later as
the tangent changes it orientation. The forward part will tend to be a
bit outside the trajectory while the rear part will tend to be inside.
This will then be accentuated by the tidal effect so that the forward
part will tend to point more and more outside of the trajectory and away from the Earth which is inside, while the
rear part will tend to point closer to the Earth center of gravity.
However, after the ship passes closest to the Earth, the tidal force
pulling more the rear part than the forward part will tend to realign
the ship on its trajectory, with the forward part in front.
I think the total effect depends on actual figures. I suspect there
may be a residual rotation speed rotating the ship as it departs from
Earth, but I have not done any calculation to be sure of it. This would imply a minute, very very ... very minute, change in Earth
rotation, because of angular momentum preservation.
If Earth is replaced by a neutron star, the intense gravity
differential between fore and aft parts may also tear the ship to pieces.