Good luck with your Alfa purchase first off - if you're new to the Alfa world then welcome also. I'd personally thank you as well as you've seen through all the scare mongering, horror stories and still decided you want an Alfa.
Firstly be re-assured that they're good, well built cars and all the electrics are Bosch (therefore German) and so chances of issues are pretty slim really.
I think most of the points have already been raised here, which is great, but as an overview:
1. The biggest emerging issue is corrosion on the sub-frame. A limited number of vehicles have started to show signs of rust on the sub-frame, particularly at the front, which is disappointing. You're likely to see some surface rust spots but some have started to rust through, so have a look into the engine bay and around the suspension arm (lower) mountings in the wheel well. If you're that way inclined some time with a set of ramps and a paintbrush could work well for rust protection, particularly around the welds. This is something to look out for but don't be too concerned about it.
2. Gearbox on the the 1.9jtdm and JTS models all share the same GM designed M32 gearbox, as features in a lot of Vauxhalls, Saabs, Alfas. In the petrol models they don't suffer as much as the torque is a lot less, but in dervs they do suffer more. It is possible for the bearing on the end of the shaft that has 1, 2, 5, 6th gears on it to fail under the load. This means a gearbox out, replace bearings rebuild on the box, which with clutch flywheel and everything else you're looking in the region of £1,200 or more. Autolusso have a great page explaining this: Autolusso - Alfa Romeo Specialist | Alfa Romeo M32 Gearbox
. As said if there is an issue this will be evident in a whining / whuring noise, possibly grinding noise from the transmission and would show play in the gear stick in 5th and 6th (which are at the end of the shaft closest to the bearing). Worth noting if you go for the 2.4jtdm or the 2.0jtdm those use the F40 gearbox (bulletproof).
3. Suspension bushes - as mentioned these are a wear and tear item and any car of any brand you should check for noises in suspension. Alfa upper arms tend to be the culprit for knocking noise - parts are about £60 each for top arms, £100 for bottom arms. Just listen for noise on bumpy / uneven surfaces, its very prominent noise if there is an issue.
4. Steering Rack - Alfa put the wrong fluid in some cars - muppets. Should be red fluid, they put the green stuff in. Problem is changing it may cause problems, so have a look and see what colour its got in there. Red would be preferable but green not always end of the world. Just check that it manages okay without excess straining noises when going lock to lock (never static steer).
The wheel alignment thing mention about is just an interesting fact. Alfa started out life with the 159 having the wheels toed in slightly on the front, which made the cars very twitchy / excitable and caused inner edge tyre wear. This was later revised to 0 so if inner edges wearing a lot you may need to get the alignment revised.
Sounds like a lot of horror stories but they are good cars and great value for money. Don't believe the MPG figures you see on statistic sites though, these are big heavy cars so will probably only return mid 30s around town and mid 40s on a run, I don't think I've every managed to get mine into the 50s.
Last note is on servicing - these are long interval cars so can be services every two years but personal experience says they really do benefit from annual servicing but that is up to you, just be aware that if it has been services every two years then it's been properly looked after, anything more is a bonus.
The cambelt if you read the book should be done at the 84k service, which if you're doing every two years means every 8 years - never leave it this long. If you read the small print at the bottom it clear states "regardless of the mileage, the timing drive belt must be changed every 4 years for heavy-duty use (cold climates, town use, long periods of idling) or at least every 5 years." Definitely make sure you follow this - exceeding it is pushing your luck.