I recently dented my Fiat Punto Active 8v 3 Door Hatchback.
I'm looking into buying this dual suction cup to remove the dent. With the severity of the damage, do you think I'd be successful with such a tool?
With the damage which is there, a suction cup dent puller is not going to do the job you are looking for it to do. If there was only "dent" damage, the suction cup can sometimes remove them. When you have creasing, pulls, dents, and paint transference, you are going to need a body shop to get this fixed.
Alternatively, you could find a replacement door at a junk yard and just replace the door yourself. There appears to be just minor damage to the door frame at the bottom. If you were to find a replacement door which is the same color as yours, replacement would be far cheaper. You'd then just need to do some touch-up work on the door frame at the bottom to get rid of the rust which seems to be forming.
Follow Paulster2's comment is to get a perfect result. But prior to going to a body shop or going to the scrapyard, you might try this:
As the dent looks really clean to me, another possibility might be to remove the inner door trims and push against the dent from the inside. You might be able to reach the dent from the inside without too much parts that are in the way. It can be dented out again with the right amount of pressure and just with your bare hands. Scrape off the damaged paint and repaint it yourself. This out-denting costs half an hour work, but might save a few bucks and a search for an equal door.