You could say that, but it is a bit old fashioned and not so natural. Nice to meet you too is a bit better

"My pleasure" is used after someone says thank you. "Thanks!" - "My pleasure!" Herman is correct. In less formal situations, you would just say "you too!" In formal, you would say "and you" or "A pleasure to meet you, too."

I would prefer answerig 'Nice to meet you, too.' Or you can answer 'You too.' for short.