My experience is from my 159 but I'm sure most of these will apply to the Brera as well:
There will likely be a lumpy idle which shouldn't be constant, from my experience anyway. It should be worse when cold and occasionally get lumpy if idling for a minute or more when warm. It might not start easily first time. The secret is to wait until the message to start the engine is displayed before pressing the engine start button. It should fire reasonably easily this way. A good engine flush and fuel additive tends to improve this, failing that a egr, throttle body and inlet manifold clean tend to sort the problem out. These symptoms can also be a sign of a failing injector/s which can be expensive to put right.
Under normal driving the engine should be smooth with a lovely, subtle 5 cylinder engine note. There is likely to be noticeable turbo lag below 1800rpm.
DPF's tend to block easily, low mpg, loads of lag and general unwillingness to rev will point to exhaust restrictions which is usually this issue. A forced regen or 2 can improve this or alternatively a dpf removal and remap reportedly does wonders for the engine. This engine is naturally free revving with good pull from 2krpm upwards. It should start to tail off around 3.5krpm
MPG will seem to be appalling until well warmed through. Water temperature should settle around 90oC from cold remarkably quickly (approx 3 to 4 miles). Average mpg on the trip computer should be over 30mpg on Trip A. You should be able to see how many miles this has been calculated over by cycling through the Trip A menu. If the mpg is low and the miles from last reset is low (less than 50 miles) it's probably nothing to worry about; on the other hand however if the mpg is low and the miles is high then be concerned. High fuel consumption may not be that simple to diagnose.
Reports of turbo failure on these engines on this forum don't seem to appear regularly so I would suggest that the turbo is likely to be sound although listen out for signs of turbo failure which is usually a high pitched siren type sound. Gearbox issues are a reasonably regular occurrence although these tend to be DMF and clutch issues unlike the 1.9 engine which also has gearbox failures.
It is likely to have suspension bushing issues on the front assembly which, depending on how many faults are found, can rack up in money. Also wheel bearings can fail.
Check the front and rear sub frames for excessive corrosion. A lot of surface rust is very likely which you would be advised to correct if you do buy as some cars are now failing MOT due to excessive corrosion. Give the under body a good going over for corrosion as Alfa tend to be hit and miss with their under body protection.
If there is no service history with the car, check with a local Alfa dealer as they should share a database and be able to see any official Alfa work that's been done to the car. they can also tell you what mileage any work has been done at so you can cross reference for 'odometer mileage correction' aka clocking. If there is no record of a water pump change within 50k miles factor in the cost of a full timing belt service with water pump which tends to be pricey.
If, like me, you like to DIY your mechanical repairs you might find the car requires quite a lot of different tools to maintain and service, so if you don't have a huge choice of mechanics tools at your disposal factor in the cost of garage costs for repairs.
Hope this helps. If it helps further here is my thread on the work I've done to my car: