Archmage of Runes's first ability doesn't change the mana cost or mana value of any spell. It changes only the total cost you pay to cast instant and sorcery spells.

Archmage of Runes's first ability can't reduce the amount of colored mana you pay for a spell. It reduces only the generic mana component of that spell.

Archmage of Runes's last ability resolves before the spell that caused it to trigger. It resolves even if that spell is countered or otherwise leaves the stack without resolving.