Killing aetherling in end step

[Bop]

During my second main phase I cast a {Shadowborn Demon}. In response my opponent exiles his {Ætherling} till the beginning of the next end step. My demon resolves and I destroy one of my opponent's other creatures with the mandatory ETB ability. My opponent is tapped out after activating the  {Ætherling}'s ability.

HERE is where I am unclear about the sequence of events.

I have a  {Doom Blade} in my hand that I want to use on his aetherling after it re-enters at the beginning of my end step. I pass priority to proceed to the end step. My opponent passes back. At the beginning of my end step the aetherlings triggered ability goes on the stack and I have an opportunity to respond as the active player. I don't have a target for my doomblade yet though because the ability hasn't resolved. I pass priority. My opponent does as well. The aetherling enters the battlefield.

HERE IS MY MAIN QUESTION:

Do I now have priority? And thus the opportunity to cast doomblade? I realize he would have the opportunity to exile it again in response, but he was tapped out. I had this come up and we played it out as though I did, and his aetherling was destroyed. Just wanted to make sure I was correct in my understanding of the sequencing of events within the end step.

[Bop]

I now realize that in the way I described it from memory I could have cast the doomblade in response to his aetherling activation before my demon resolved and circumvented this timing issue as he would have been unable to activate the ability again in response. I'm not sure whether this was the exact sequence of events, there may have been other spells cast that eventually had him tapped out as we entered the end step.

[Pleeb]

I want to say section 503 deals with the end step.

513.2.: Second, the active player gets priority. Players may cast spells and activate abilitie

I'm interpreting this to mean that players can cast instants during any end step, not just those that have triggers.

[Kaleo42]

You must pass priority and then have your opponent pass priority to end the turn. After {ætherling} comes back it is your priority again as active player. Only when both players are totally content doing nothing will the turn end, discard to max handsize, and remove damage.

RESOLVED