So after a couple of hours of playing around and testing I've finally found the correct settings to achieve this, I wasn't able to find the configuration options through the Windows Airport Utility client but borrow a MacBook Air from a family member and was able to find it:
1. Open the AirPort Utility on a Mac computer.
2. Click on your AirPort Device, then click "Edit".
3. Select the "Internet" tab.
4. Now in the "Domain Name" section type "yourhostname.ddns.net" or it could be "yourhostname.no-ip.com" or whatever other hostname No-IP.com may have assigned you.
5. Now click the "Internet Options" button.
6. At the bottom of the windows that opened tick the check box "Use dynamic global hostname"
7. In the "Hostname" section fill it with "no-ip.ddns"
8. In the "User" & "Password" sections fill it with the username and password that you chose when creating your "No-IP.com" account.
9. Now click "Save" which will save the settings and click "Update" to update the Airport device which will also restart it.
And you're done!
Please note that it does sometimes take up to 5 to 10 minutes to update the DNS settings to No-IP.com, which caused me to think it wasn't working initially.
Good luck!