As far as I can tell there's not that much to go badly wrong with these.
When was the cambelt changed since it's recommended every 3 years or 36000 miles.
Listen out for creaking or squeaking from the front suspension when you drive (often after it's warmed up a bit) which could indicate worn wishbones (bushes or ball joint).
Also, take a look underneath the car for corrosion. I had that on mine and didn't realise until I'd handed over the cash and got home.
It's generally accepted that the protection underneath the car isn't great out of the factory. But then again mine is a lot older than yours so corrosion not likely to be an issue.
Take a look for uneven wear on the rear tyres especially. Again, a common fault that could indicate worn rear suspension bushes or incorrect tracking (to be honest I'm not sure, not looked into the cause of mine but it's a common issue).
Also, check if the airbag light is staying on. Common issue and easily fixed.
That's what I would look out for having bought a 147 1.6 a few weeks ago.
Some of the more experienced Alfa Owners will I'm sure give you some further advice.
Plus these forums and its members are an absolute goldmine of information on how to keep your Alfa in good health.