First, the use of "got" is confusing you. It's not standard English and is not usually taught in text books. Many people will say it's bad English. Change it to "have" and maybe the sentence is easier to understand? "You and me have a lot of catching up to do." Does that help you? Now think about grammar, not vocabulary. What does "we have a lot of X to do" mean? How about "we have to do a lot of X"? You should be able to figure it out now. You just need to know what "catching up" means. Maybe you can make a guess here, and I'll let you know if you got it right. (Hint: you can also say this with a reflexive verb - "catching up with each other")

It means you and the other person need to tell each other what you have been up to recently, and what may have changed since you last spoke to each other. If you haven't seen someone in five years, and they now have a wife and children, you might say, "Wow, we have a lot of catching up to do!" And then, presumably, they would tell you how they met their wife, when they got married, and about their children, and other changes in their life since you last spoke. Another example: You may have "a lot of catching up to do," in school if you missed lessons, or homework. In this context, it means you have to learn what you missed since you were last in class. I hope this helps. If you have more questions, I'd be happy to help you.