I did this on my last 159, the badge was a faded as, needed changing.
The "wings" stay on the car, it is only the centre Alfa badge that is changed.
Using a drill, make a small hole in the one on the car, use a screw driver to wiggle the old one off. When you have a decent sized piece sticking out, use pliers to rip the badge off. It is only made of tin with a laquer outside. Peels off really easily, fixed on with double sided tape.
The 166 badge is mounted on a metal backing, with two "prongs" sticking out. These need to be eased off with a pair of pliers. The 166 badge already has a ring of tape fitted to the outer edge of the reverse.
When the face of the part on the car is cleaned, offer the badge up to check the fitting. The 166 badge sits slightly proud of the plastic chrome of the original 159 badge, but I think it is fine. When happy, remove the sticky tape backing, and press firmly.
Job done, time taken, 5-10 mins. Cost, as cheap as you can find a 166 badge.