Runic Armasaur's ability doesn't trigger when an opponent activates an ability of a card in hand (such as a cycling ability from the Amonkhet block) or a card in a graveyard (such as that of Bone Dragon), even if that causes a card to be put onto the battlefield.

Activated abilities contain a colon. They're generally written "[Cost]: [Effect]." Some keyword abilities are activated abilities and will have colons in their reminder text. An activated mana ability is one that produces mana as it resolves, not one that costs mana to activate.

Runic Armasaur's ability resolves before the ability that caused it to trigger. Players can cast spells and activate abilities after the triggered ability resolves but before the activated ability that caused it to trigger does.

Runic Armasaur's ability doesn't trigger when an opponent activates an ability of a noncreature, nonland permanent that causes it to become a creature (such as a crew ability of a Vehicle).