I'm not sure the kit is cheaper. And I'd be doubtful of its benefit, and without seeing what you're looking at, I don't know its quality.
Look here:
https://www.shop4parts.co.uk/?name=...pts=1100&term=Fiat_500_1.2_8v_Shock_Absorbers
These people are specialists, and well respected on here. If you
donate to the forum, it opens up access to a discount from them too. I've selected 1.2, so ensure you look at the correct parts for your car.
You will also see listed some bolts and other fittings. If yours need replacing, you need to use the correct ones, as they are stronger than 'normal' bolts. Bolts are graded, standard are 8.8, but some stuff availabloe from hardware stores are lower than this. Suspension bolts are 10.9. Even using 8.8 is a risk.
The 500 front suspension is good, but does wear. Wallowing does suggest the dampers are past their best, but everything needs to be checked.
The links between the suspension and the anti-roll bar fail regularly, usually causing a clonk, and often wear can only be seen after the part is removed. They are simple and cheap.
https://www.shop4parts.co.uk/?name=store&op=Product&ProdID=2266
The lower suspension arm bushes, especially the rear ones, do perish, and get soft. The rear bushes can be replaced, but this requires a press, and they need to be aligned properly. Complete arms are easier, and renew both bushes, and the ball joint. Always replace as a pair, both sides.
The top mounts also get soft.
https://www.shop4parts.co.uk/?name=...=1100&term=Fiat_500_1.2_8v_Bushes_/_Mountings Make sure they are fitted the correct way around.
Coilovers are likely to be harder than original, to try to show some difference, and would be sold as a 'benefit'. That makes the ride harsher, and transmits more forces to the car body.
Have a very good look at all the front suspension. And replace all that is needed with standard equivalents. Keep it stock, but good, the manufacturer knows best.