As for the circuit, I don't think there is much of circuit for this, it would just connect across your power rails, much like conventional decoupling, with the following reservations... The initial charging of the cap pulls a lot of current from the battery. Plus, the switch for the circuit needs to be after, the battery/cap combination, you don't want to have to charge the big cap every time you switch on, if its discharged through your circuit.
One way to overcome this problem, fit everything 'Except' the battery, then 'pre-charge' the cap from an external PSU/bigger battery to the correct voltage, 5.5V? when the cap volts, fit the batteries. Note if your using 6V batts? you will need a diode dropper arrangement, to ensure that the 5,5V 1F cap doesn't get more than 5.5V (I think most can handle 5.7V peak)
2nd way, use a current limiting resistor to charge the cap, this avoids the heavy inrush current. The circuit then runs off the cap, via a switch.
Note I've haven't tried this yet, as my application is direct solar powered, and doesn't have this problem.
Maybe some other expert here may have a more precise solution? These caps are a bit new technology, and are usually feed with a bit more power than LR44 batts. But I think the theory is reasonably sound.
Steve