I don't run (in fact, I have a bad hip that keeps me from doing it). I raced in 2 cross races this season. I've done fine. Not last is good enough for me. It's a blast. The amount of running/being off the bike will vary by course. I was 'running' (more like a goofy walk/jog) more on the last course because of a sand pit and a long 'run-up'.

I'd recommend working on cardio fitness as prep. The whole getting off the bike and running or some facsimile thereof spikes my HR. The point of puke was always just at the edge of my presence.

As for bikes... Cross bikes can be had for ~1200 MSRP. I have a specialized tricross sport somewhere in that price range. I have seen women racing on Mtn bikes though. So if you think you might like it, you could always try a race with an MTB. You'd be slow, but you'd at least get an idea of the sport.