Can an electron at rest be set in motion by the magnetic field?

Sure. A charge in a magnetic field experiences a force.

A force acting on a mass produces acceleration.

Remember the old-style TV sets that were about 6 feet deep from front to back ?

That type of picture tube is called a 'CRT', for "cathode-ray tube'.

There's a hot wire at the back end of the picture tube that produces a cloud of

electrons. The front face of the picture tube has a high positiver voltage on it,

to attract the negative electrons to the front. On the way there, the electrons

have to go through these magnet coils that are around the neck of the picture

tube. The magnetic field inside the neck pulls the stream of electrons left and right,

up and down, left and right, and that's how they draw a picture on the front face

when they get there.