Officina Milano is certainly one of my favourite places for a night out in Milan. It's also among one of the the World's 50 best bars, so you can expect great cocktail here.
It's my go to bar, it has some of the best cocktails I've ever had in the city and the location is just amazing. You won't find many places as beautiful as this one in Milan.
The interior features vintage motorcycles, cars from the 1950s, leather sofas and atmosphere is just great. I consider it one of the best spots for a night out in Milan.
Cocktails are around 12-15€, the average for most cocktails in Milan. They also have a restaurant and you can also have an aperitivo here.
I highly recommend to book a table in advance, as especially in the weekends is all booked.