Do "at the start of your turn" abilities work due to a swap with Alarm-o-Bot?

Short answer is no. I like to reference Magic the Gathering (MTG) in regards to this, because Hearthstone basically uses the stack like MTG, but it's rarely relevant, only with situations like this.

So the explanation for this is that, at the start of your turn, the game checks and queues up all 'start of turn' effects on the battle field, putting them on the stack based on the order the cards entered the battlefield. Then, the game goes through one by one and resolves each of the effects, not checking for any new effects. Then the turn progresses to the next step; drawing a card.

If you think about the situation you describe, if the game was coded like that and if you had an Alarm-o-bot in play, and the only minion in your hand was another Alarm-o-bot, the two Alarm-o-bots would try and infinitely swap with each other, completely breaking the game. So, unfortunately for you, your doomsayer will still take a turn to activate.

See the third answer by the Hearthstone Community Manager HERE for confirmation.