Do temporary units like Sprites die before the "when I hold" triggers of battlefields like Startipped peak?

Yes — Temporary units are killed in the Beginning Step, which happens before the Scoring Step where Hold points and "when I hold here" triggers (like Startipped Peak's "channel 1 rune exhausted") occur.

Temporary is defined as "kill this at the start of your Beginning Phase, before scoring" (Rule 816.1.b), and hold effects only trigger at a battlefield you actually Hold and score in the Scoring Step (Rules 315.2.b, 383.4.d, 471.2). So the Sprite is already gone before Startipped Peak's ability could ever fire.

Practical consequence: if the Sprite was your only unit there, its death queues a Cleanup in which you lose control of the now-empty battlefield (Rules 190.4.c, 323.6), so you never Hold it — no point, no trigger. To get the Hold trigger you must react to the Temporary trigger (hidden flip, Ambush, Sprite Call) so a unit is present when the Scoring Step arrives, or already have a non-Temporary unit there.