No tools -the rod goes into the hole in the plastic bush( white thingy!) which then you rotate and clip the end tangs over the rod itself which locks it in place so it dont drop out when you operate the handle like you are now experiencing!.
Dont know if you can buy the little bush seperately might be worth ringing a Fiat dealers parts department to find out! if they do sell em you can bet you have to buy a packet of 10 or the like doubt they sell them individually! try an e- bay search as well!.
When putting back together go SLOW have a good torch to hand so you can see properly inside the door card and dont rush think it through and you will be ok look at the bush before you re-fit it and you will see how it works with the tangs horizontal and the hole for the rod vertical!
good luck!
ps there is a seller on e-bay DG RALLY who sells a set of 4 clips two white/ two blue -that fit alfas and fiats for around £6 inc postage
I searched FIAT 500 WHITE DOOR ROD CLIPS and it came up!