Play style
The play style to take advantage of the Ambassador is something like a behind-enemy-lines sniper. The best way to think about is is as a long-range back stab, since it requires precision to use effectively, but when you are successful, you will take down enemies very quickly.
Ambassador = alternatives
Per usual, your goal as a Spy is to sneak behind your enemies. Normally, your next goal, apart from sapping the Engineer's structures, would be to get in position to back stab your enemies, but with the Ambassador, you have the option of trying to get in a few quick head shots instead, which lets you attack your enemies from further behind than normal, allowing you to play more cautiously than normal as well.
Ambassador vs Revolver
Since the Ambassador emphasizes head shots, and since head shots are easier to get on enemies who are unaware of your presence, the Ambassador is more suited for initiating conflict in situations where a back stab attempt is unfeasible or risky, such as when you come across a sniper equipped with the Razorback, or a Pyro. It would be better thought of as a weapon suited for 'back-of-the-head' shots rather than one simply suited for head shots.
The basic Revolver is more suited to ending conflict by finishing off survivors of failed back stab attempts, or holding off enemies who are pursuing you. Also, since it is normally more powerful than the Ambassador, it is also better for finishing off the Engineer's structures, especially after you have begin sapping them.
Summary
If you are successful at sneaking behind enemy lines but have difficulty in getting up close for back stabs, the Ambassador can be an effective alternative.
Outside of an uber-charge, if you manage to reach a sentry gun before it can turn to face you, you can circle-strafe it... that is to closely run around it in a circle while facing and attacking it.
This works best as Pyro and Scout, although a Your Eternal Reward Spy with no disguise can uncloak and circle-strafe a sentry to plant a Sapper on it as well.
Note: You have to watch which way the sentry is turning and circle it in the same direction.
Note 2: It's also important to note that circle-strafing fails if the sentry is flush against a wall. However, since it's much harder for an engineer to hide behind a sentry that's against a wall, most Engineers won't build them that way.
Best Answer
I tested with the Gas Jockey item set. Kills with the Degreaser, Powerjack, or my secondary weapon which isn't part of the item set, all generated calling cards, for every kill. So it looks like whenever you kill or appear to have killed someone while you have the item set in your loadout, a calling card is generated. There is no catch.