Should I get $200 for passing "Go" if I then get a card that says "don't collect $200"?
Yes, they would still gain $200 from passing GO then landing on a square that makes them draw a card like this.

You collect $200 immediately when you land on or pass GO, then the player would continue their move and land on Chance/Community Chest, then draw that card. 

The text on the card specifies that you cannot collect $200 if you pass GO (again) in the process of moving around the board from a Chance/Community Chest square to the Jail square.