Hi Karen,
I can't help much on apartments downtown. But I can suggest this site for lots of ride information:
www.bikepptc.org
Go to the ride schedule button and you'll find lots of ride info. The good thing about DC versus central NY is you can ride through a lot of the winter here. Better than spending hours indoors on a trainer or even in a spinning class. Many of the rides originate in DC or close by you can ride to the start. In fact one ride I regular do Saturday mornings in the winter that starts at Belle Haven marina has several DC people that ride to the beginning.
You can also get some good deals on gyms in your situation where I assume you'll be using them primarily in the evenings and weekends. Many of the downtown clubs cater to day workers at noontime, morning or early evenings. They really thin out in the evenings and offer discounts to locals that use them in non-prime time.
Stan