McCrae is leaving the FTSE 100 pharma giant at the end of March and will join the postal operator, which floated on the stock market four months ago, on 1 April.
He has been recruited as a deputy for Royal Mail’s MD of strategy and communications Shane O’Riordain, whose remit was expanded in January.
O’Riordain took additional responsibility for pricing, regulation and government affairs from Sue Whalley, who was promoted to chief operating officer.
He has made the deputy role responsible for media relations, internal communications, public affairs, corporate responsibility and community investment.
"Neil brings a wealth of communications experience that he will draw upon in this busy and challenging role," O'Riordain said. "Royal Mail is a major UK business, employing 150,000 people across the UK and operating in every community."
McCrae has spent six years at AstraZeneca, in jobs including vice-president of corporate communications and head of global media relations.
He joined in 2008 from rival GlaxoSmithKline (GSK), where he was vice-president of UK corporate affairs and European communications. Before joining GSK predecessor SmithKline Beecham in 1996 he part-owned a communications agency called Design Publications.
Esra Erkal-Paler, head of global media relations at AstraZeneca, said: "We wish Neil all the best in his new role."
She said she was not currently in a position to say how the company planned to replace him or whether there had been any change in the role of interim executive vice-president of human resources and corporate affairs Caroline Hempstead.