Since nobody answered this question yet.... The cheappest way is $6 for the 5A metro bus that will take you downtown (L'Enfant Plaza stop) then probably $2 or so to your hotel on the metro. The 5A runs about every 40 minutes, I think.

There's also the "Washington Flyer" bus that goes to a metro station closer to IAD. I believe it's about $10-15 for the bus.

Even if you're a cheapskate like me , it's at least worth a $20 price difference to fly into DCA. it's also a tiny airport, so you'll be in and out of there quickly. Last DCA flight I took, plane pulled into the gate at 9 am, I waited for my checked luggage, took a taxi that got stuck in morning traffic, and still got to Dupont Circle by 9:30.