The tune is Seán Ó Duibhir a[n] Ghleanna (John O'Dwyer of the Glen). It has been discussed here a number of times under those names, and you can find notation at J C's Tunefinder by searching for John Dwyer of the Glens; John O'Dwyer of the Glen; Ghleanna; or Duibhir. You won't get the modern piano intro, though!

The Battle of Augrim is a completely different tune.