Yes, that's the inner joint. Did you rotate the wheel to look for hidden splits?
Kind of impossible to describe what it should feel like. But compare it to the other side - if it's obviously looser then that's the problem. Probably.
I've never done an Alfa one - cars haven't lasted long enough yet... But they're normally pretty easy to change. I did one on my Renault camper a while ago and it was a bit awkward because it was held in by a roll pin, but still only took about an hour and a bit by the side of the road.
Normal procedure is: Jack up, support and remove road wheel. Loosen hub nut (assistant's foot on brake) and tap inwards with hammer to loosen shaft, then remove nut. Split bottom balljoint (on 156 you might have to take the bolt out which holds the strut too) and pop outer end of shaft out. Inner end is normally held with a snap ring - shock it out with a lever and remove. Catch oil. Replace shaft and top up gearbox oil.
Someone here will tell you how to do it properly.
Might be worth undoing the cover and refilling the joint with grease. Molybdenum disulphide grease will do. You'll need a new clip or big cable tie to put it back on. Sometimes they've had a rubber boot replaced after all the grease has fallen out and putting new grease in quietens it down. Did that to the big diesel Merc I had when it got noisy and it lasted another 5 years.
I'd caution that it could be something else, although it definitely sounds to me as though it's that joint. And if you're not confident take it to the garage, because you don't want the car falling on your head.