If I have an instant that gives a creature +1/+7 (Aegis of the Heavens) until end of turn, I should be able to cast this after my opponent declares their attackers, right? Because the game hasn't been giving me the opportunity.
Yes, declaring attackers happens during the Declare Attackers step (rule 508). After declaring attackers (508.1), the active player (your opponent) gets priority (508.2) but you'll get priority thereafter and should be able to cast an instant.
Unless there are other effects in place barring you from casting spells, it seems to be an Arena bug.