First, panels that you can use...
There are a few at Adafruit (here, here or here)
You just want to check that the barrel jack and plug are the same size. I couldn't find dimensions in the documentation for the energy shield so you may need to dig a little deeper or just swap it for one that you know will work.
Also, the question about how long it will take also depends. First, if we assume that the arduino is not running we can get ~500mA from the second solar panel listed. If we can get all of that into the battery, it would take (1200mAH/500mA) about 2.4 hours to fully charge the battery. It looks like you have to make some adjustments on the shield to get the most out of your solar panel. And of course, you'll want to adjust those calculations based on the panel that you end up getting.
Now, if we assume the Arduino is running and consuming about 25mA you will have to subtract that 25mA from the 500mA that the solar panel is providing. So, in general...
(Battery Capacity)/(Solar Panel Current - Arduino Current) = Charge
And you have a lot of control over the solar panel current and the arduino current based on the exact arduino that you use, etc, etc.
There is also probably some loss regulating the solar panel voltage from 5-6V down to charge the battery (which is probably at 3.7V). I also don't know if the charging circuitry will limit the amount of current going to the battery.
Take Away: There's a lot of stuff to consider when trying to answer this question. I think your best bet would be to get a panel that fits and dump the fuel gauge values out to the console every few minutes.