Complete the following sentences:

Would you like to come to lunch with Noah and ___?

Just between you and ___ …

When the word in question is the object of a preposition and not the subject of a sentence or phrase, we should use the object pronoun.

A subject pronoun is the performer of the action in a sentence.

The object pronoun is the receiver of the action in a sentence.

So the correct answers are:

Would you like to come to lunch with Noah and me?

Just between you and me…

I is a subject pronoun and me is an object pronoun. 

I like chocolate. / The call is for me.

So a good way to check yourself is to remove the other person.

And just one more important thing:

It is considered polite to mention oneself last in double subjects or objects.

Why don’t you and I go away for the weekend?

The invitation was for Tracy and me.