I have a laptop, a Toshiba C55-A. Bottom of the barrel specs, as well as the dreaded AMD integrated GPU, Windows 8.1, 4 GB RAM. It runs TS12 SP1HF4 pretty well. It hiccups a little, but no big deal. I don't know how well it would run T:ANE because I don't have it, but if you want cheap, I would say something a little better than what I have. You won't have very smooth frame rates, and you will have to tamper with the graphics settings quite a bit, but it should be playable.
Matt