I have no specific idea aboot the 1.3 but Fiat fuel pumps are basically all built the same... so have a general look to see what makes sense to your pump;
The pump is connected to the rest of the system by fuel pipes that terminate in push-fit plastic plugs, that have an o-ring around their base to make the seal.
The pipes are push-fit but they are usually secured place by a metal collar, or yoke that is itself held down by a screw or bolt.
One thing that can go wrong is that the plastic push-fitting plug can fret against something.. either a metal piece of the pump, or the yoke that's holding it in place. Since it's nylon/plastic, it will eventually wear through and this will cause a leak.
The other thing that can go wrong is that the pressure in the lines can push out the plug slightly and unseat the o-ring. Or the o-ring will shrink/age/harden just a bit. These will both result in a leak.
So... take the covers off the pump, get some degreaser or some detergent (kicthen Mr Muscle etc. will do if you don't have any pukka vehicle/engine dfegreaser).. and a paint-brush (or use your missus' toothbrush) and clean the pump and pump area to buggeration, in order to get it clean and dry.
Then start the engine to see what's leaking and from where.
The plastic connector plugs only come bonded to a whole new replacement fuel line, so if you see any damage on yours, you'll have to get round the breakers yards (or try Fiat for a new one, if they're not too £shocking).
O-rings you might have more luck.. but if not/or as a temporary fix you can get some Blue Hylomar or instant gasket type goo (it has to be fuel resistant) and smear it around the o-ring before carefully sticking it back in the receiver. Be careful to not to get the goo into the sharp end of the fuel line, or it'll block something expensive, like an injectore.
Ralf S.
