When I'm in Kyoto, I always stay at the Hearton, which is fairly near the area you specify. Right at the corner of Kawaramachi and Oike, the two major subway lines cross right below. There are stations for both at the intersection. Nijo-jo and the Imperial Palace are in walking distance. So is Pontocho. You don't mention what size room you are looking for but the double-bed room is in your price range. The 2-twin room is a little more pricey but not too bad. Bus stops for the major tourist buses are a couple of blocks away in either direction but it is also really simple to get the combined bus/subway pass and just take the subway to start each day at Kyoto Station if you get confused.

http://www.japanhotel.net/Hotel.aspx?ID=18