Though primarily known as a poet, William Butler Yeats wrote nearly a dozen plays, as well, including At the Hawk's Well and Deirdre. As a close friend of Synge's and fellow founder of the Abbey Theatre, Yeats's theatrical works are certainly worth studying in relation to Playboy of the Western World.
Samuel Beckett is another famous Irish playwright, and his explorations of absurdism in theatre may be found in works like Waiting for Godot and Endgame.
Works by Sean O'Casey surrounding the Dublin slums include Juno and the Paycock, The Plough and the Stars, and The Shadow of a Gunman.