When I went to the Washington Humane Society on New York Ave, I was able to adopt a cat then and there. But make sure to read their steps for adoption first (http://support.washhumane.org/site/PageServer?pagename=adopt) – I brought along a copy of my lease (along with an addendum from the landlord verifying that I had permission to have a cat) and a completed adoption form. They didn’t actually sit me down for an interview or anything, just chatted informally with me briefly.
You can also prepare by looking online at the cats and dogs they currently have at the shelter – but I’m not sure how frequently they update the database, and it’s really not the same as in-person interaction.
Best of luck finding a new friend!