There are a few different calculations you can do that will help figure this out.
Energy consumption
The first is how much power you'll need to run the system. Each LED has a voltage drop of 2.0 V. Assuming you run the LEDs at their maximum current (20 mA), they'll each use 2.0 V * 0.020 A = 0.040 W. For 12 LEDs, that's 480 mW, assuming the rest of your system is perfectly efficient. In an hour, you would use 480 mWh, or 0.480 Wh.
As a first estimate, let's say your system will be only 50% efficient, so you need 2 * 0.480 Wh (roughly 1 Wh) to run for an hour. (You can probably do better, but we'll get to that.) A CR1225 coin cell is a 3 V battery rated for 47 mAh. That's the same as 3 V * 0.047 Ah = 0.141 Wh. That's about 1/7th of what you need, so your LEDs will last around 1/7th of an hour, i.e. 8 or 9 minutes, with one CR1225 coin cell.
Efficiency
Can you do better than just guessing 50% efficiency? Sure.
It depends on how you wire the LEDs and what current-limiting resistors you use. You could make 6 strings of 2 LEDs each and run them off of a 6 V pack made from two coin cells in series. Each string would have a 4 V drop, so you'd need to drop the remaining 6 - 4 = 2 V over a resistor. At 20 mA per string, that's a 2 V / 0.020 A = 100 ohm resistor.
In this case, 1/3 of your power would be dissipated in the resistors (since 2 V is 1/3 of 6 V), so your efficiency would be 66%.
You might try 4 strings of 3 LEDs with no ballast resistors, but I suspect that their brightness would drop off quickly as the battery voltage sagged. Your efficiency would be 100%, but the brightness would suffer.
You could also try running off of the 5 V supply you mentioned; with the right resistor, you could hit 80% efficiency.
I have used the Seeeduino Stalker with Waterproof Solar Kit successfully. I think this kit is a great place to start as it will work out of the box. It consists of an Arduino compatible board with LiPo charger, solar panel input, SD card, real time clock, and water resistant case.
The wireless sensor node kit you suggest is a standalone board for XBees, it would not go with the UNO.
The solar panel and charger should work with the UNO. The battery is the same one as in the Stalker kit.
By the way, the Arduino UNO has an inefficient regulator with about 10mA quiescent current so it consumes a fair bit of power just sitting idle. This can be an issue with solar powered devices. The Stalker has a more efficient regulator. You may also want to have a look at sleeping the Arduino to save more power.
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Best Answer
The Arduino Uno isn't fit to run off a coin cell.
The Uno isn't exactly low-power. It contains 2 microcontrollers running at high clock frequencies and consumes a couple tens of mA
A coin cell will give you 3 V, while the Uno is designed to run on 5 V.
Most important: a coin cell has enough energy to power the Arduino for a couple of hours, but can only supply this in small doses, i.e. a few mA. That won't work.