Well luckily, there are no artifacts or creatures that have such an additional cost, so this situation shouldn't ever come up.
But to answer the question, let's assume there is such a creature/artifact as stated.
When casting a spell you do everything in order:
1)Announce the spell and move that card from where it is to the stack. (Rule - 601.2a)
2)If the spell is modal, select the modes. (Rule - 601.2b)
3)If the spell targets, targets are chosen. (Rule - 601.2c)
4)If the spell divides anything (i.e. counters or damage), the division is announced. (Rule - 601.2d)
5)The game checks to see if the spell can be legally cast. (Rule - 601.2e)
6)The player determines the total cost of the spell. (Rule - 601.2f)
7)If the spell costs mana, mana abilities are activated. (Rule - 601.2g)
8)The total cost of the spell is paid in any order. (Rule - 601.2h)
9)Once the above is done, the spell is cast. (Rule - 601.2i)
Paying costs for a spell is the last step that must be done when casting something. So by the time you sacrifice Hostage Taker, the spell has moved from exile and is on the stack. And has legally been cast.
So the creature/artifact you stole with Hostage Taker is now on the stack, and when it resolves will be yours, in this hypothetical situation.
But to answer the question, let's assume there is such a creature/artifact as stated.
When casting a spell you do everything in order:
1)Announce the spell and move that card from where it is to the stack. (Rule - 601.2a)
2)If the spell is modal, select the modes. (Rule - 601.2b)
3)If the spell targets, targets are chosen. (Rule - 601.2c)
4)If the spell divides anything (i.e. counters or damage), the division is announced. (Rule - 601.2d)
5)The game checks to see if the spell can be legally cast. (Rule - 601.2e)
6)The player determines the total cost of the spell. (Rule - 601.2f)
7)If the spell costs mana, mana abilities are activated. (Rule - 601.2g)
8)The total cost of the spell is paid in any order. (Rule - 601.2h)
9)Once the above is done, the spell is cast. (Rule - 601.2i)
Paying costs for a spell is the last step that must be done when casting something. So by the time you sacrifice Hostage Taker, the spell has moved from exile and is on the stack. And has legally been cast.
So the creature/artifact you stole with Hostage Taker is now on the stack, and when it resolves will be yours, in this hypothetical situation.
Hope that helps.
Moderator note: With the rules question answered, I am locking this thread. Bringing up examples is good, but we don't need to list them all. -MadMage