I have my Pi hooked to a monitor and a USB hub that connects a keyboard and a Phone with USB Tethering turned on
The Pi Zero W auto-detected the new internet connection and set it up as usb0
You can verify it with a lfconfig command and that will also give you the IP Address to log into the AstroPrint side after you have done all of the raspian updates.
sudo apt-get update
sudo apt-get dist-upgrade
sudo apt-get install rpi-update
Use the browser on the phone to access that IP Address from usb0 as it is an internal address assigned by the phone. Use the phone to setup the wireless in AstroPrint.
I also updated AstroPrint while tethered but I am not sure I needed to.