Can electricity be made by passing a magnet through a coil instead of spinning it?

You need a conductor and a magnetic field. Any relative motion or change in field strength generates an electric current. Go to youtube for demonstrations of these things. Like this:
www.youtube.com/results?search_query=magnet+in+copper+tube

The moving magnet generates a current in the copper (generator effect) and that opposes the motion of the magnet (motor effect).

Here is some really deep science: www.thunderbolts.info/wp/daily-tpod/