As noted, the airports are pretty far apart.
If you do plan to use the Metro as noted above, leave yourself plenty of time. It's not the most reliable system and there are lots of breakdowns and delays.
A taxi would probably run somewhere around $75-80 or so.
As for where to stay, Crystal City is not a bad option as there are many hotels there so Priceline or Hotwire is a decent idea. There are a few hotels out near IAD if you'd prefer to take your time in the morning.