Is it possible to have some sort of a spaceship that basically runs on electricity and "butt kicks" itself forward? Or would it stand still because every force has an equal force in the opposite direction?
It would stand still, because of the equal and opposite force rule (aka conservation of momentum).
On earth's surface it's possible to have a self-propelled butt-kicking machine by having a slow "wind-up" phase alternating with an abrupt "kick". During the wind-up, static friction with a ground surface holds the vehicle in place, but the kick phase can break the frictional threshold. However, the momentum of the system of (ground + vehicle) remains conserved, ultimately using the Earth's vast mass as a "momentum sink".
In space, with no ground to brace against, the butt-kicking machine goes nowhere.
If you used 1.000.000 cubic meters of steel to create a 1 mm² wire spanning a large distance of 1000 million km you might be able to tug spacecraft along it.
Not very practical though and the wire might be unstable or snap. IF it worked and you could get something like 70 000 N out of it (~70 kg on Earth) that would be quite good though.
You would always need to have spacecraft going both ways to keep it stable and each craft would have to share the max load with all the others.
Depending on what you count as a 'mechanical' you could consider Project Orion. (Technically a nuke uses nuclear energy, not mechanical but it would be 'pushed' by the explosion)
Basically, detonate a nuke and use it to propel you. So a very hot kick in the butt.