Really, I need more information but will help as well as possible. The price range is possible if you already have a rear wheel with a cassette/driver type mount for your rear gear. If not, forget it.
The rear wheel will cost you a little over half of your budget for a very cheap wheel and up to the total amount if you want a good one.
Let's say you go the cheap way: The 3 pc crank will cost you at least $60-120 depending, again, what quality you want. The sprocket will cost you between $25-$50, again, depending on what quality you want. And finally, the 9t cassette is pretty cheap at between $10-$30 depending on material it is made out of... and quality.
You will also have to make sure that your hub is compatible with the cassette/driver as some companies like Profile and Chris King make cassettes only compatible with their own products.
So... here is what I'd recommend: Go to a store that sells Mirraco bike (that is Dave Mirra's brand.) Ask them to show you a bike with a "9t driver" for the rear gear. They should be able to show you some bikes that are a little more than what your budget is but you will get a great quality bike with a lifetime warranty without having to hassle around with changing parts out and things not being compatible. Trust me, I have changed out these new parts on old bikes in my shop and when everything is said and done, I have had to always order extra parts to make the old bike work and that cost the customer a lot more money than what they initially thought it would.
Sorry to be the bearer of bad news.
