Not really :) I had a busy Sunday / Last Sunday was busy / I had a busy day last Sunday would all work.

As others have explained, it doesn't sound right to us. I'll try to explain why that is. When I hear "I had a busy last Sunday", the question that pops into my mind is "A busy what?" Was it a busy "day"? Was it a busy "schedule"? Whatever it was that was busy deserves to be modified by "busy" by putting "busy" directly in front of it: "I had a busy day last Sunday".

I was busy last Sunday. I had a busy day last Sunday.

Last Sunday was busy