"Blazer Gill Rabus" has two different triggers when he enters the field. The controller of Blazer places the triggers in the order of their choice after he enters the field, before either player has priority. Players can respond to both triggers or in between the triggers, after the first one has resolved.
Blazer's first automatic ability does not prevent the opponent's J-Ruler from gaining new abilities after it resolves. If the opponent of Blazer gives their J-Ruler Imperishable after the first automatic ability resolves but before the second resolves, their J-Ruler will have Imperishable when it dies.