So, in case there are others out there who are as hopeless as I am and would like some help, this was the solution:
The 2009 Passat radio is connected to two permanently live wires coming from the battery.
My older radio requires one permanently live wire (to remember the settings and stations), and one wire that comes on with the ignition (to switch on and off when the car is). In the past I have learned that sometimes these wires need to be swapped around, but one needs to be always live and one switched with the ignition. In the case case of the 2009 Passat, this is not the case.
So, to swap your old radio for a new one, you need to remove the old one. There are plenty of videos on the internet that show you how to do this, so I will skip that step. You will then need to put in a fascia and some kind of double DIN adapter. I bought mine at a regular car place. Again, there is plenty of help on the internet about this, so I will assume this has been done.
Once the radio is out, you will have a connector sticking out of your dashboard that looks like this:
Meanwhile, your other car stereo will have a connector that looks like this:
These need an adapter to fit together. I bought mine from a regular car place off the shelf:
Note that several of the wires, including the red one, come with bullet connectors for easy disconnection and reconnection.
Next we have to locate the fuse box (I realise there are better tools than a screwdriver to open this, but it was what I had to hand):
Get out a volt meter, as we need to find an empty fuse slot that is not live when the key is not in the ignition, but becomes live when you insert the key.
In my case, the top left fuse slot fit the bill. It is a mini-fuse.
Next, we need to run a wire from the fuse box to the stereo. I bought the wire from a regular car place again: