Charging a lead-acid battery in a car
It's complicated. 

If you connect just a diode to this UB battery, and the radio to the UB battery, the UB battery would indeed keep the radio memory when you disconnected the main car battery. 

But there's a problem here: when the main car battery is connected, and the engine is running and the alternator is supplying energy to car's system, and if the alternator has a voltage 0.6V (this if for the diode voltage drop) higher than the battery maximum, the alternator would keep feeding energy to the UB battery, and maybe wouldn't be safe keep feeding UB battery after it's full charged (especially this chinese battery on the photo; I wouldn't rely that it would deny the extra charging). If this battery explodes, remember that your car has a reservatory full of petrol...

Automotive batteries are designed to keep high voltage, in a few minutes they get to the alternator voltage and so they stop beeing charged. A few minutes ago I answered another question saying exactly that automotive and deep-cycle batteries are so different.

If you use a passive componente to limit the UB battery charging, doing the "trickle charging", this wouldn't help, too: if the engine is on, and the radio is on, the radio would "eat" more energy and the UB battery would be empty very soon. 

I think that would be safer if, besides the diode to keep the UB battery to feed back the entire system, there were some control circuitry to watch the UB battery voltage and stop the charging if it is full charged.

EDIT: Maybe would be simpler and safer if you keep two direct wires (can be thin wires) from the automotive battery to the radio memory circuit. Put a fuse in the positive wire, and, after connecting those wires, disconnect the radio from the vehicle main power and from the negative chassis (from now on the radio will be powered for those wires).

Doing this way, you don't need to add this new battery to your vehicle, but keeps the radio memory even when the battery is not connected. It could work, unless the radio's memory drains so much battery (I already heard about hungry car's stereos even in standby mode).