I've got this assignment and it states that we can't use any methods that involve creating a library of cDNA clones, but is based on RT-PCR (Reverse Transcription – Polymerase Chain Reaction).

The only thing we know is the mature protein sequence.

How do I design a primer that is specific to the mRNA?

Well, I don't think you could design a specific primer because of the genetic code redundancy. BUT you could design a collection of primers based on the amino acid sequence allowing for each of the possible amino acid codes. However, you would have a TON of primers, and you might also lose hybridization specificity.