My blue Nasus dies in combat, but manages to kill an enemy unit. Do I still channel a Rune Exhausted?

No. You do not channel a rune exhausted.

Nasus, Guardian of Knowledge's ability reads "Once each turn, when an enemy unit here dies, channel 1 rune exhausted." When Nasus and an enemy unit both take lethal combat damage simultaneously, they are killed together during the same cleanup step (Rule 323.5). Per Rule 383.2.c.2, a triggered ability cannot evaluate its condition if its source leaves the zone it's active from at the same time the trigger condition is satisfied — the example given is Viktor, Leader who similarly does not trigger when he and another unit die together in the same cleanup. Since Nasus leaves the board at the exact same moment the enemy unit dies, his ability never fires.