Will Intel i3 2120's APU run League of legends on high settings?

Don't understand... If you mean Intel i3 2120's integrated graphics, then no.

If you get the a10-5800k, then yes. You will get playable frame rates. It's similar to a descreet 6670, just slightly worse performance.

Maybe at low resolution like below 1366x768.

"APU" is a term AMD came up with for CPUs with integrated graphics. The A10's might be able to do high settings at low resolution.

Intel still calls their unified CPU + graphics chips CPUs.

i3 2120 has HD2000 graphic, atrocious long before it even came out for anything 3D

either add a video card to go with it, or get the newest and fastest AMD A-series