Is it still possible to access the AirDroid by going to a specific address?

I do not want to install AirDroid as WifiHotspot. I prefer the "old" way of being a web-server listening on 8888 port if I remember well. Is that still possible?

Do you mean Lite Mode? If your Android and computer are on the same network, you still can use AirDroid via Lite Mode. More details: http://help.airdroid.com/customer/portal/articles/1296277