I'd look for a single stage Honda. Getting it in that price point may be difficult but this time of the year it may be possible. However, if it hasn't been maintained, expect to replace the carb if it has a lumpy idle, the paddles and scraper bar (normally all replaced as a set). That should run you about $100 and if you're reasonably handy, you can DIY it. So now it's $100 over budget, but you'll have a very stout little thrower.
Couple things to note:
1. It's meant for lighter duty work, but it's very capable. Things it can do - move up to 6 inches of heavier snow and 12" of lighter snow. Note that the heavier the snow, the shorter the throw, down it as little as 8 feet or so (in contrast to light snow getting throw 30 feet). Things it won't do (well) - the stuff the plow leaves at the end of the driveway or re-throw the snow a second time. Bear this in mind if you're dealing with heavy snow, you get one shot to throw it, if you have to throw it 15 feet, this isn't going to work.
2. In terms of the area you need to clear, it only matters with really large spaces. It only has a 20" width, so that can translate to more time outside clearing the snow compared to something with a 30" width.
3. With heavier snows, you'll need to go out multiple times during a storm to clear so you're staying within the capabilities of a single stage machine. If you can't do that, think carefully about whether a single stage is right for you Once you get beyond the boundaries outlined above, things get worse fast. You could spend 2 hours with a bit of frustration peppered in, trying to clear your driveway with a single stage where a 2 stage would knock it out in 30 mins.
I use a single stage 90% of the time, since we typically get lighter snows. The maneuverability and lighter weight make it very convenient in use. But point #3 above led me to get a 2 stage for those bigger events. I have no desire to be out there for 2+ hours screwing around with an undersized blower.
