OK, thanks. I am assuming that the "main" AirPort connects to the Fios router now, and that the other AirPorts connect to the "main" AirPort.
If that is the case, then things should work if you connect the AirPort Extreme to the new Altice One router as follows below. Please note that the word "should" is not the same as the word "will" though.
Power off all of the AirPorts in any order that you want
Power off the Altice One router
Change the Ethernet cable that connects to the main AirPort from its current connection at the Fios router to one of the four yellow Ethernet ports on the Altice one router
Power up the Altice one router and let it run for a minute or two
Power up only the main AirPort and check to see that it has a green light and that devices can connect to the AirPort network
If all is OK, power up the second AirPort and run the same checks again
If OK, power up the third AirPort and run the checks
If OK, power up the fourth AriPort and run the checks
If OK, you are all set.