How does Sheltered Aerie work?
No, all players' mana pools get emptied at the end of every step and phase of the turn.


  500.4. When a step or phase ends, any unused mana left in a player’s mana pool empties. This turn-based action doesn’t use the stack.


There are some cards  that let you keep some or all of your mana, but they always state that clearly in the card text, and Sheltered Aerie is not one of those cards.