Yes you can, and both sentences are fine. Here's why:
As well as the regular uses of the verb 'do', it also has a specialist use as what is called a pro-form. As a pro-form (in this case a pro-verb) 'do' substitutes a verb or verb phrase:
'Fred walked to town and I did too'. (where did replaces wal
Basically, the purpose of a so-called pro-form is to avoid repetition. In this respect, Avangi's post hits the nail on the head for your second sentence, Alex.
Secondly, the intended focus of "not only" is the direct object in the sentence ("you" vs "me"), not the verb. If we eliminate the inversion and reword accordingly, I think you will see how completely unnecessary it is to repeat