If my opponent has Vex at a battlefield and I have a hidden card there, do I have to pay the additional cost to cast that spell?

Yes, you must pay the additional cost.

According to the official ruling for this interaction:

• Hidden sets the base cost of the spell to 0.
• Vex's ability acts as a cost increase, which is applied after the base cost is set to 0.
• Therefore, you must pay the additional cost imposed by Vex to activate the spell from Hidden.

Sequence:

1. Hidden removes the base cost of the spell.
2. Additional costs (such as the cost increase from Vex) are added after.
3. You must pay these additional costs to activate the spell.