Will it be cheaper to buy an Ball joint or go to an mechanic to let him buy it?

You can buy it yourself cheaper... but here is the trick. You buy and most mechanics will just charge you more for doing the work. They buy and you pay less for labor, so in the end the cost will be about the same. Best thing to do, let the mechanic buy it and if he gets the wrong one or has a problem, he deals with it... you buy and there is a problem, you also get to deal with it.