If Blue Nasus and another unit trade on a Showdown, does the Blue Nasus trigger and channel 1 Rune Exhausted?

No — Nasus's ability does not trigger.

Nasus, Guardian of Knowledge's trigger is "once each turn, when an enemy unit here dies." When he and the enemy unit both take lethal damage, they're killed together in the same combat cleanup step (Rule 323.5). Since Nasus leaves the board at the exact same instant the enemy unit dies, he can't evaluate his own trigger condition — Rule 383.2.c.2 says a triggered ability fails if its source leaves the zone it's active from at the same time the condition is met (same as the Viktor, Leader example). He has to be alive when the enemy dies. (FAQ #11803)