Boot iOS directly into Airplane Mode
You can't do that. You can, however, make it so your phone does not have internet access when it boots.


Remove your iPhone's Sim Card (you can get a sim card remover on Amazon for maybe $3).
Scan for wifi and ensure no wifi is reachable in case the device is paired with a network. 
Disable any hotspots (Bluetooth or wifi-based)
Boot up your phone normally, and it will have no method to connect to the internet.


Note that Bluetooth and gps will still be active since you won't be in full airplane mode, so this probably isn't suitable for a plane flight.

Optional: To ensure no pre-existing wifi or Bluetooth connections, perform a factory reset of your phone via iTunes or Apple Configurator or other tool.