What would happen if a spaceship would pass near a completely stationary astronomical object? Imagine  a completely stationary astronomical object (e.g a planet) and assume it has no movements of any kind (no rotation, no orbits around a star or even a galaxy... a completely still planet)
If a ship passed near the planet and began suffering its gravitational pull, could it use that gravitational force to change its course and be accelerated (even if just for a little bit)?
 A: Yes, you're right. It would accelerate a tiny amount.
A: One can always take the system of the planet at rest to use the gravitational formulas for the trajectory of an object  in the gravitational field of the planet, which should be simple as you have "no rotation, no orbits around a star or even a galaxy" no other gravitational fields except the two, spaceship  and planet.
There are several things that can happen depending on the trajectory of the spaceship, the sure one is that there will be an attractive force between the mass of the planet and the mass of the satellite which will accelerate the spaceship towards the planet. Depending on the masses and the velocity vector of the spaceship the trajectory would be a conic section. If it is a parabola or a hyperbola the spaceship will leave the vicinity of the planet without need to change velocity. If it is an ellipse (or circle) it will be caught around the planet.
You ask:

if a ship passed near the planet and began suffering its gravitational pull, could it use that gravitational force to change its course and be accelerated (even if just for a little bit)?

Not true.  By the nature of the gravitational force, the gravitational acceleration is built in the trajectory the spaceship finds itself in . To change from an ellipse to a hyperbola, acceleration must come from the spaceship, in order to avoid being bound or going on a crash direction ( the ellipse so narrow that it hits the planet).
A: Technically correct but not what you wanted answer: yes, both the ship and planet will accelerate. Acceleration is any change of speed, which is not what you ment.
You likely ment if velocity of a ship can increase or not. For this you need to define the reference point - what frame do you use to measure velocity?
You probably ment the planet as a reference plane. Then again, we can find technically right, but not what you ment answer: ship will accelerate as it comes closer to the planet.
So, how would a question sound, to avoid these non-answers above? Probably it will involve total energy of the ship - sum of potential and kinetic energy. So that when the ship falls towards the planet, that doesnt increase the value you ask about, because that would be a boring answer.
With the last question an answer is simple - no. Energy can not change. It can not increase or decrease. Which means when ship will fly far away fron the planet - it must have exactly the same velocity as it started with, if distance became the same. Ignoring speeds close to the speed of light for now.
With 2 bodies of similar mass situation is more complex, center of mass have to be taken as a reference frame, but result is the same, energy is conserved.
With 3 bodies it is different. One body can in fact 'take away' energy of other 2 bodies, and fly away faster than it started with, at the same distance.
This allows to make a simple rule of thumb: gravitational maneurves are useful only if frame of reference changes.
Hint: no need for a 'still' planet. All of the above works with any planet, in orbit or not. Well, almost, orbital motion is not straight, but this detail is miniscule for most gravitational maneuvres.
Suggestion: check the kerbal space program. Gravitational maneurves are major part of the gameplay. Interface with them is a bit hostile, but still provides lots of insights.
