1981 Fiat 124 Spider won't stay running!

[JcL]

I bought this fiat from an owner that had it since 1984, It was a good friend of mine's mother, she said it used to run great before she went to college when she had it sitting then forgot about the car and parked it at her dad's house. It was sitting for over 14 years! I changed the fuel pump, injectors, fuel filter, fuel lines, all electronics seem to work, fuses arent blown or rusted. Overall the car seems great! It has rust here and there and the interior is PERFECT.

Now for my problem...
The car cranks then starts for no longer than a second then dies.
Sometimes it doesnt even turn on.
Could someone please tell me what else I can do?
Is it reccomended to change distributor cap and rotor? They seem perfect inside. Everything seems brand new when I open it though.
I haven't changed the spark plug cables either so I don't know if that could effect that much either.

Please give me ideas! I would love to get this thing back on the road asap! That way my friends mom could see her childhood back to life