I also query why Alice has "ua Briain" attached to her name, the de Burgh Irish connection started with her son William de Burgh. I suspect Alice was born in Norfolk, and can't think of any evidence linking her to Ireland.
The shield that is shown for her appears to have first been adopted by the de Burghs in the later 12th century and has no connection to her. The earliest de Burgh shield for which there is any evidence is lozengy gules and vair (and variants) (see Ellis page 229). See also the pommel found in Burgh-next-Aylsham that has a shield with lozengy gules engraved on it, it might have belonged to one of the de Burghs.
Is there any evidence that Alice died while on the "road to Palestine"? The only evidence that I'm aware of that Alice even existed is the one mention concerning the advowson of the church of Oulton (see Ellis page 191, "Alice, has come to light in one document").
Stephen