Wow, never heard of an APU but after reading up on them, it sounds a great way to save money. So this APU, do you think it would run LoL fairly well?
I couldn't find any particular benchmarks for LoL, but I don't think it's too demanding. The A8 APU can even do Skyrim if on Low settings and that's a fairly demanding game, so I'd guess medium settings at least. I would definitely say get an A8 if you just use the integrated, it has a better GPU. It will be at least about as good as the 440.
I mentioned a "decent graphics upgrade later", these APUs when combined with a few certain discrete cards can be crossfired for a 2 GPU configuration. The HD6670 when used with an A8 makes a 6690D2 combo, and has been reported to play Skyrim up to high settings, so should be enough for any "normal" gaming. It's also fairly affordable (about $70-100) and wouldn't need a larger PSU, should you feel gaming is lacking later on.
It's always nice to keep a low power draw, especially on a media PC that will be on a lot.