The cassette will work with your existing wheels. You will need a new shifter and maybe derailleur. I think Shimano 10spd derailleurs use a different pull ratio than 9. But it may be road derailleurs that are like that.

You are looking at probably ~$200 to gain one gear. The biggest gain on 10spd was that they went to an 11-36 cassette. Do they make 11-36 nine speed cassettes? That may be a better option.