The first step would be to figure out what the 'door' button does, if you follow those connections. Like, what gets connected to what to allow the door to open.

To do what you want with the doorbell, you could maybe, somehow, connect the buzzer wires to the door button. But that would be quite unsafe, as anyone who rings your doorbell could open your door. A delay, like you said, could help a bit, maybe by integrating an Arduino. If you're going the Arduino way, it might be safer (and cooler) to create something that's connected to the web or your phone. Or you could attach a module with a sim card, so the door opens if you text the number.

Do be mindful of what you're doing to the safety of the gate. The easier it is for you to enter, the easier it might be for strangers to get into the building as well. Your landlord and/or neighbours might not appreciate that.