When I have a temporary unit that channels runes on death, do those runes channel in the beginning phase and then awaken so that I can use them that turn?

No, that is not how the timing works. The Awaken Phase happens at the very start of your turn, before the Beginning Phase where your Temporary unit triggers and dies.

The Turn Sequence

1. Awaken Phase: You ready all your exhausted game objects (Rule 315.1.a). This happens before anything else.
2. Beginning Phase: Your Temporary unit’s ability triggers, resolving to kill it. This triggers its Deathknell ability.
3. Resolution: Because the Deathknell ability is resolving during the Beginning Phase (after the Awaken Phase has already concluded), the runes you channel enter the board during the Beginning Phase.

Because you have already passed the Awaken Phase, those newly channeled runes will enter the board (typically exhausted) and will not ready until the Awaken Phase of your next turn. You cannot use them to pay costs during the current turn unless you have another effect that specifically readies them.

References:

• Rule 315.1.a: The Awaken Phase, where you ready your game objects, occurs at the very start of your turn.
• Rule 315.2: The Beginning Phase occurs after the Awaken Phase.
• RiftJudge FAQ #8926: Confirms that runes channeled during the Beginning Phase remain exhausted until the Awaken Phase of the next turn.