This one man play, directed by Polly Findlay, sees Krapp sitting alone on his 69th birthday listening to a tape recording he made 30 years earlier.
The production opens in the Crucible Lyceum Studio on 1 July 2014 (previews from 25 June), and runs until 19 July.
Tickets are on sale to general public from 8 February.
Wilson, an associate director at Sheffield Theatres, is best know for his portrayal of Victor Meldrew in One Foot in the Grave.
His numerous stage credits include Twelfth Night (West End and RSC), Whipping It Up (Bush Theatre, West End and national tour), Waiting For Godot (Royal Exchange Theatre Manchester) and What The Butler Saw (National Theatre).
Polly Findlay returns to Sheffield Theatres to direct – she previously worked for the company on A Taste of Honey. Her other recent credits include Protest Song (NT Shed), Proof (Menier Chocolate Factory) and Antigone (National Theatre).