I think you'll find that the biggest difference between the two chains is how WIDE the link is, and the skinnier chain won't fit over your cruiser sprocket. I recommend you get an inexpensive chain tool at a hardware store and use it to take a section out of your OLD chain and add it to your NEW 1-3 speed chain.
The tool works easily, and once you've practiced a bit on the old chain pushing the pins out and back in I think you'll be okay.
TRICK: push the pin out almost all the way, then flex the chain sideways to open it. If the new chain comes with a master link you should splice the links in a way that keeps the master link in the chain. It's handy when you need to open the chain.