lillii:
The LED strip will be something like this one:
http://www.amazon.co.uk/Lighting-EVER-Flexible-Waterproof-Strips/dp/B004JRK35W/ref=sr_1_8?ie=UTF8&qid=1352245506&sr=8-8The sections (i.e individual arduino outputs) will be around 1m each (0.4A) but there may be one that is around 2m or slightly longer.
Yeah, so each 1 meter will need 400mA. You can get dual mosfets that can drive 1.5A (each), or even 1A ones, in a package no larger than 3mm x 2mm. Just saying. 
Since you said you won't be lighting all the strings up at the same time, battery consumption will be whatever you are pulling for the strings, so 400mA per string, plus a bit of overhead for the Arduino (clone or not.) Regular AA batteries can handle roughly 2A per hour. So flipping 400mA through various strings will get you approximately 5 hours worth on 3-4 AA batteries. It gets better with a LiPo, however you will need to build a regulator circuit so the LiPo's voltage can be brought down for the Atmel. I'm willing to bet you can run those strings with a 3S cell battery. The only thing is, 11.7V is a long way to drop to 5V for the Atmel and you will develop some heat doing that. Might be worth running the strings off of the Lipo, and the Arduino clone off of 4x AA batteries. Sure it's two packs, but you won't be dealing with a whole lot of heat doing that.
I have been known to attach a custom PCB right onto the 4x AA battery pack, with three wires going to a string of LEDs (DATA, CLK, GND) and a heavier battery pack elsewhere supplying the strings (the GND from the Arduino clone ties in with the big battery pack's GND.)