I'm not much of a microbrew beer pub guy, so others might need to back this up or refute it, but there is an Astro Beer Hall about 5-6 blocks from the Arena. I've only been to a different one just south of the city, but they had a ton of good microbrews.
Astro was originally just a chicken and doughnut place founded by former Cap (and current Lightning Assistant Coach) Jeff Halpern.
If for whatever reason you ended up in "Old Town Alexandria", "Bugsy's Pizza Restaurant and Sports Bar" is a Caps bar that was founded by former NHL player Bryan Watson who played 877 career games, ending with the Caps in 1979. I'll assume you can Google to find specifics.
Much like
@Kalopsia said - I can't help too much with hotels in the city. I only ever pay attention to them for large conferences I attend. They're generally pretty expensive for "good" ones.
The areas between Georgetown, "West End", and George Washington University have a lot of good places to check out, things to do, fun little bars. Maybe bonus (?) is that the Swedish Embassy is near that area.
I'll assume you have/can check out the resources the Embassy may have, as far as any travel advice, things to do, etc.
Borrowing the same map Kalopsia included, I added two black circles on the right side. Those areas you might find cheap lodging, but I would think twice before staying there. Obviously, this is very general, not perfect, and I may include an area that's fine. The city has improved a lot over the past 15-20 years, but there are still areas that might not be a lot of fun. Basically, areas to the east of the Anacostia River might be worth avoiding.
Last thought I have is that I took some extended British and Australian family, who had never been in DC before, to the International Spy Museum. They said it was the best part of the trip. It's a pretty cool place, if you have any interest in that kind of thing or in history of World War 1 and 2.
