There was a very good reason why Alfa fitted 16" alloys on the Veloce(sport models) as opposed to standard 15"alloys..... Why not go for 17"'s or 18"s, they could have quiet easily done that?
But I digress from your question
The 1.8 TS is probably the best balanced engine as it is what is called a "square block" meaning it is neither long or short stroke.
As far as engine tuning goes, well there isn't much you can do with a limited budget apart from helping it breath easier.
A decent cone filter and a good quality full exhaust will set you back about £350+ and around an extra 5-6 horses maximum which will put you around the same as a 2.0TS at the very best.
Chipping is a waste of time on a TS you may be lucky to get an extra 8-10 horses but you will suffer for it in other ways... Leave the remaps to JTD's
You can go further with uprated cams and cylinder head work but the price jumps up considerably and then it simply isn't viable as changing the car to a 2.5V6 24v is far more cost effective....
Or you can go crazy and either supercharge or turbo the motor... A big risk on an older engine and ££££'s to get installed.
Instead of thinking "power" start to look at it from a different approach... Handling & brakes!
A well sorted 156 will show the way home on the twisty stuff to 300+bhp cars...Honest!
A quicker point to point car is the way to go... You can hold the speed up on cornering a lot better and brake later, its almost phenomenal how the 156 chassis performs
The standard opinion of combo's varies from owner to owner but the general consensus is
Koni FSD's matched with Eibach springs and Eibach uprated A/R bars is the best bang for buck.
Either Brembo, Black diamond, Tarox, Ferrodo or EBC brake pads and discs offer much better abilities in stopping and bleeding off speed.
Now this brings me back to my original opening comment...
Wheels! I have had 15's, 16's,17's and 18's and you know what, for the cost effectiveness of tyre choice per size... A well shod set of 16" alloys offer a much better compromise of handling, poise balance and comfort than the rest of them.... You will find that out for yourself soon enough.
If it was my wallet, I would put the 1.8 TS up for sale and buy a V6 with a broken cambelt for a couple of hundred quid , dump the engine and put a 166 3.0 V6 lump in with minor mods , A Q2 diff, Remap and then uprate the suspension and brakes... You will have something near to GTA performance for a couple of grand
PS. Speak to Alfa tuner on what is possible!