Can I sue the actual client who name was text ed to me, who happens to be a celebrity?

That is why they built small claims courts. You will probably find: 1) any judgment in your favor will be appealed to the Superior Court where attorneys are allowed. 2) obtaining a judgment will be easier than collecting the judgment.

Somebody owes you for your time and services. Typically in a lawsuit, all potentially responsible parties are named. I would think you'd name the person who "hired" you as well as the passenger you transported.

No, you can't sue the celebrity. Celebrities are immune from all lawsuits. (Just kidding.) More than likely, this is a matter that the limo company you work for and the person that ordered their services should resolve. Leave it to your employer to resolve. If you are not being paid by the limo company, your claims would generally be against them, not the customer.