Is it possible to power up Arduino upon serial transmission without dropping data?
I never tried myself, but afaik you can define an wake up/interrupt from USART, meaning that if you get a receiving byte from the UART it will wakeup the Arduino.

When powered up, you can read continuous data. I would be surprised if they make an UART wakeup signal, that bytes would be lost (i.e. that powering up would take too long to miss more UART bytes).