Connecting to internet using access point

Am I able to unplug the AirPort Extreme from the modem and plug direct into one of the ether ports on the access point box?

Normally yes, (see the info at the bottom) but in order to so so, you will need to reset the AirPort Extreme using a Hard Reset or Factory Default Reset and then set it up again while it is connected to the access point.

https://support.apple.com/en-us/HT201945

After the reset, connect an Ethernet cable from the access point to the "O" port on the AirPort Extreme and then set it up again, just like you did when the AirPort Extreme was new. You can use the same Base Station name, same WiFi Network name and Password that you used before, or you can set up new names and password.

Most companies prohibit employees from installing their personal devices on the company network, and some company IT departments set up the equipment to not allow other devices to connect unless you have specific administrator passwords. It's your call to decide how you want to handle this.