Do temporary units die before I'm able to activate Dusk Rose Lab ability?

No, temporary units do not die before you are able to activate the ability of Dusk Rose Lab.

Because both the Temporary keyword (Rule 816.1.c) and the ability of Dusk Rose Lab trigger at the start of your Beginning Phase, you have the opportunity to resolve the Lab's ability first.

Sequence of Events

1. Beginning Phase starts: Both the Temporary trigger (which aims to kill your temporary unit) and the Dusk Rose Lab trigger are added to the chain.
2. Chain Resolution: As the controller of both triggers, you choose the order in which they resolve. You choose to resolve the Dusk Rose Lab ability first.
3. Cost Payment: You kill your temporary unit as the cost for the Lab's ability.
4. Effect: You draw 1 card.
5. Temporary Trigger: The Temporary trigger then attempts to resolve. Since the temporary unit is no longer on the board (it was already killed to pay for the Lab), the instruction to kill it cannot be followed and is ignored.

This interaction is confirmed by RiftJudge FAQ #9231, which explicitly states that you can use the ability of Dusk Rose Lab to kill a temporary unit before it is removed by its Temporary trigger.