I am running my old version of FCP (6.06) on my MBA and have been very pleased with its performance. I previously ran it on a 2008 15" MBP (c2d 2.4ghz with 4gb RAM) and upgraded to a 2011 13" i5/4gb/256gb MBA. This was a big improvement for me - that machine had a cpu almost 2x faster than my old machine, but the SSD made the overall experience much, much faster.
This summer I upgraded to a 2013 11" i7/8gb/512gb MBA and it runs even better. Rendering is noticeably faster. Aside from that though, it feels pretty much the same.
If you can afford it, get the maximum spec like I did. If not, I think you will still be fine with the i5 and a smaller SSD. I would get 8gb RAM either way, although. The 512gb SSD is blazing fast though - I get about 700mb/s read and write performance. The 128gb SSD is much, much slower. Not sure about the 256.
Caveat: like I said, this is an old version of FCP, not sure how newer versions would work. But I suspect you will be fine. The MBA is a really powerful little machine.