You can't take the bonus-action attack before the Attack action, because it is conditional on you taking that action.
Rules designer Jeremy Crawford addressed the same general issue in a tweet on May 11, 2018 (prompted by the similar wording of another feat):
Clarification about bonus actions: if a feature says you can do X as a bonus action if you do Y, you must do Y before you can do X. For Shield Master, that means the bonus action must come after the Attack action. You decide when it happens afterward that turn.
This ruling on Shield Master and similarly worded feats/features was then formalized in the 2019 Sage Advice Compendium, which mentions that "the same sort of if-then setup" appears in other rules:
The Shield Master feat lets you shove someone as a bonus action if you take the Attack action. Can you take that bonus action before the Attack action?
No. The bonus action provided by the Shield Master feat has a pre-condition: that you take the Attack action on your turn. Intending to take that action isn’t sufficient; you must actually take it before you can take the bonus action. During your turn, you do get to decide when to take the bonus action after you’ve taken the Attack action.
This sort of if-then setup appears in many of the game’s rules. The “if” must be satisfied before the “then” comes into play.
(Notably, the first page of the 2019 Sage Advice Compendium also indicated that now, none of Crawford's tweets are considered official rulings. Since I've noted it here, I'll avoid reiterating their unofficial nature in the rest of the answer.)
Crossbow Expert's third bullet point is worded in the same fundamental way as the first benefit of Shield Master. Since the bonus-action attack granted by Crossbow Expert is predicated on you using the Attack action (specifically attacking with a one-handed weapon as part of it), then by the same logic as above, you can't take that bonus-action attack until after you have actually taken the Attack action.
As Crawford explained in another tweet from May 25, 2018:
D&D combat doesn't have an action-declaration phase. Things happen in order, and you can be interrupted at any moment by a reaction, trap, or the like. You can say, "I plan to take the Attack action," but that has no rules relevance until you're actually taking the action.
This revised ruling contradicts a previous tweet by him in January 2015, in which he stated that, "As with most bonus actions, you choose the timing, so the Shield Master shove can come before or after the Attack action." Crawford acknowledged this change in this tweet from May 11, 2018:
In 2017, I changed the ruling on bonus action timing because the old ruling was illogical. The original ruling failed to account for the fact that X relying on Y is a form of timing. The new ruling corrects that oversight.
And further explained his reasoning in another tweet later the same day:
The old ruling on bonus action timing didn't quiet questions on that timing. Instead, the illogical ruling fueled questions, and it even inadvertently led some fans to think our choice of words like "if" or "when" had super-precise meanings in bonus actions. They don't.
Since the bonus-action attack from the Crossbow Expert feat is reliant on taking the Attack action and making an attack with a one-handed weapon as part of it, that bonus-action attack can't be made until you've already made an attack with a one-handed weapon as part of the Attack action.
Notably, if you're able to make multiple attacks as part of the Attack action, it might be possible to interrupt the Attack action to make Crossbow Expert's bonus-action attack, since it's reliant specifically on attacking with a one-handed weapon as part of the Attack action - rather than on the Attack action as a whole. However, that should probably be asked as a separate question if it's unclear.