Any 【Bonus】 on a unit lasts until the start of that unit's next phase. If Effective against dragons is acquired during the enemy phase, it will disappear when the next player phase begins.
The effective damage applied by Effective against dragons does not stack with existing effective damage. However, a unit must neutralize all sources of effectiveness to avoid effective damage. For example, if Slaying Spear+ is also effective against a unit, that unit must neutralize "effective against armored" and "effective against dragons".