Can I play Cull the Weak to kill the opponent's unit without any unit present on my base or at battlefield?

Yes, you can play Cull the Weak even if you have no units in play.

According to the RiftJudge FAQs and the game's rules:

• No Units Required: Cull the Weak does not target specific units when it is played. Because it does not require a target to be finalized on the chain, you can play it regardless of your board state.
• "Do As Much As You Can": In Riftbound, if an instruction cannot be fully followed, you perform as much of the effect as possible (Rule 359.3.e.11). If you have no units, you simply do not kill one, but your opponent must still kill one of theirs if they have any.
• Not a Cost: Killing a unit is part of the spell's effect, not an additional cost to play the card. If it were a cost, you would be required to have a unit to sacrifice in order to play it.