THE "PRINCIPAL" PROBLEM IS THAT I HAVE NO TIME
This sentence is actually correct and can be used in English but, at least in verbal form, I prefer to use the word "main". So the sentence would be something like: "The main problem is that I have no time".
In addition, to this you need to be careful with the different uses of "principal" and "principle" both of which are pronounced the same but have different meanings. "Principal" is often used in the context of a school when you want to talk about the director of the school, for example. "Principle" on the other hand, is connected to the idea of morals: "It is a matter of principle".
Hope that helps!
Soy el director de Daniel Smith + Partners. Tengo más de 10 años de experiencia enseñando inglés a españoles.