One pretty simple thing to do is to run a current through it and watch it twitch. Lay it out horizontally on a smooth, hard, nonconducting surface, stretch it just a little so there are small spaces between the turns, and briefly make wire connections from the ends of the slinky to the terminals of a battery. It should visibly contract. I don't know the resistance of your slinky, so I don't know what voltage you should try, but I'd start with a single cell, preferably big like a C-cell or a D-cell, and just make a momentary connection. You want to get a current pulse of 5 Amps or so. If the single cell is too weak, try 2 or 3 in series. Since the current in each turn of the coil goes in the same direction, the turns attract each other (see ref. 1).
Here's another, but it isn't electrical. Listen to your slinky cry its heart out. See ref. 2. Since you have the small version, you'll need to figure out some way to couple it to the cup, but some glue might help.