I doubt that you can run Final Cut Pro on these things, and even if you do, most likely it would run ├╝ber slow. Netbooks are not meant for high-end graphics work.
Possibly, you could install it with an external monitor installed so it gets through the resolution check, but otherwise, I don't think you'll have much luck.
Macbook (April 2008) | Intel Core 2 Duo 2.4 GHz | 4 GB RAM | 250 GB 5400rpm HD | OS X 10.7.3
Dell Mini 10v (August 2009) | Intel Atom N270 1.6 GHz | 2 GB RAM | 120 GB 5400rpm HD | OS X 10.6.8 (borked) / Windows 8 Consumer Preview
GA-X58-USB3 (January 2011) | Intel i7-950 3.2 GHz (OC'd) | 6 GB RAM | 1 TB 7200rpm HD / 120 GB 5400rpm HD | Windows 7 Ultimate / Windows 8 Consumer Preview / OS X 10.7.2