I am always amused at this kind of approach to immortality. While the copy of me that is reborn would appreciate my preparedness, that doesn't make this copy any less unhappy about dying.
It is possible one day you will go to sleep biologically and wake up non-biologically. It’s just a matter of sufficiently emulating the processes that were present when you were the biological you.
I try to be good to the guy who wakes up with my memories and body the next morning. That makes me no less unhappy about dying.
So there you have it :)
The idea of a continuous self is probably an illusion, but if your body dies that illusion does too. Your copy just lives its own version of that illusion.
But it is not as if the brain destroys and creates brain cells at anywhere close to that rate. The neurons and the connections between them are quite long-lasting. I'm going to assume that's the "me" in here, until proven otherwise.
The process would probably involved disconnecting parts of the brain while simultaneously activating the emulated versions. It might be possible to do while awake even. but not for a long time