Can I use full the weak when I have no units and my opponent has 1?

Yes, you can play Cull the Weak even if you control no units and your opponent controls one.

According to the official rulings for this card:

• Targeting: Cull the Weak does not target. Because it says "Each player kills one of their units," the choice of which unit to kill is made upon resolution, not when the spell is played.
• Partial Resolution: Under Rule 356.3.e.11, instructions that can be partially followed are followed as much as possible. Since you have no units, you simply ignore the instruction to kill one of your own units.
• Opponent's Requirement: Your inability to kill a unit does not prevent the rest of the spell from resolving. Your opponent must still kill one of their units if they have any.

Sequence:

1. You play Cull the Weak (it does not require you to have a unit to be legally played).
2. The spell resolves.
3. You have no units to kill, so you do nothing for your part of the effect.
4. Your opponent must choose and kill one of their units.