Can I sue a person that is slandering my name throughout the company? How?

Perhaps, but I would have to have more facts to form a firm opinion. Generally anyone can bring a suit, the question is whether or not you can win it and recover your costs and provable damages. Slander cases are difficult because proving the damages is not easy unless it is a slander per se. You should take all of the facts and circumstances to an attorney for a recent opinion.

Yes, you may sue. To win, you will have to show actual monetary damages. Judges and juries do not favor slander suits, so it is unlikely you will win much, if anything. I suggest you consult with a local attorney. A better solution may be to discuss this with the employer's human resources department.