There are still a lot of downtown hotel rooms available. It's well worth staying downtown and walking as much as possible. On Innauguration Day, most of downtown will be closed to all but official traffic, it will be hard to cross many streets other than at designated places and many Metro stops will be closed.

I would simply check with event sites to make sure that they will be open. Also, remember that the innaugural is on Sunday and that Monday is MLK Day, so a holiday for most, and everything will be back to normal by then.

The key is to wear layered clothing to deal with changing weather and have things like water & snacks with you. Pay attention to security rules about what you can and cannot carry where, and all will be well.