The closest hotel to Union Station, as some have said and anyone can ascertain, is the Phoenix Park Hotel. According to Google maps, it is 0.2 miles from Union Station and the time it takes to walk there from Union Station is 4 minutes. Obviously, depending on your pace, mobility, etc. you may get there quicker or slower than average. Hence the walk may be anywhere from 3-6 minutes for most. If I recall correctly, most of the walk to the hotel is downhill, which is nice if you have luggage. Obviously, upon returning to the station it will be uphill (it is a modest grade, not a hill really.)
I have stayed there twice off Amtrak LD trains. I can attest that it is a nice hotel quite suitable for business and tourists too. I believe it is Irish and has a pub. I've eaten breakfast there and it was quite good. As some have said, the rooms tend to be small, in the European tradition. Decor is in the same tradition.
As for rates, I would say it's mid-range, but people's wallets vary in size so it's best to check out rates on your own. For walking to and from Union Station with no public or private transportation, I don't think you can do better. (The Kimpton George Hotel is one minute further from the station, per Google Maps.)