Yes, the clicky starter fix will fit on your samurai, no matter what engine it has in it. All the fix does is take the wire that goes to the starter solenoid and uses it to trigger a relay that activates the starter. The relay fix is pretty simple to make, using a common 30 amp relay and some wire and some crimp on terminals.
Sounds like to me, that a clicky starter issue may not be your problem though. Generally, when a clicky starter is the problem, you will hear the starter solenoid click each time you turn the key to start....which I didn't see you mention.
What I would do first is clean the positive battery cable connections that go to the starter. Clean it at the battery and where it attaches to the starter solenoid. Use sandpaper to polish up the connections, until they are bright and shiny. Then use some dielectric grease on them, when you reassemble.
If you want to use your old samurai starter on your 1.6, you can (even though they are not the same starters, they will interchange). I don't really recommend it though...because the 1.6 is a better and more powerful starter. Also, if you use the 1.3 starter, you may have to grind a little on the side of the 1.6 block, for the starter to fit (because the samurai starter is fatter than the 1.6 starter). Although, when I used a 1.3 starter on a 1.6 it fit fine without having to grind the block.
