The couple were welcomed by schoolchildren from St Joseph's National school in Kilmessan, Co Meath, waving Irish and Monaco flags.
Albert signed the visitors' book before a full welcome ceremony including a 21-gun salute by the 2nd Field Artillery Regiment, from McKee Barracks.
The national anthems of both Ireland and Monaco were played to mark the start of the visit.
The prince inspected the guard of honour, made up of Defence Forces' soldiers from the Army's First Southern Brigade, along with Naval Service and Air Corps personnel.
Ms Wittstock stood with President McAleese during the ceremony, wearing a three-quarter length black coat and beige trouser suit.
Government officials have said the visit will reflect the long-standing ties between Ireland and Monaco and underline the potential for future co-operation in business, environment, research, education, tourism and culture.
The prince was introduced to the Irish reception party which included Taoiseach Enda Kenny, Ireland's ambassador to Monaco Paul Kavanagh, Garda Commissioner Martin Callinan and Defence Forces' Chief of Staff Lieutenant General Sean McCann.
He also met the children from St Joseph's, before attending a private meeting with President McAleese, her husband, Ms Wittstock, Mr Kavanagh, and Monaco's honorary counsel in Dublin Howard Kilroy.
Prince Albert then planted an Irish oak in the garden of the Aras, accompanied by President McAleese and the school children.
He left for his next engagement at the Garden of Remembrance in central Dublin with a motorcycle escort.
Security was tight around the Garden of Remembrance, a memorial garden in Dublin's north inner city which honours the memory of all those who gave their lives for Irish freedom.
Albert was greeted by Defence Minister Alan Shatter and walked the length of the garden past the cross-shaped water feature with Lt Gen McCann. His fiancee followed, escorted by Brigadier General Dennis Murphy.
The prince stumbled slightly as he walked up the steps towards the large sculpture based on the theme of the Children of Lir.
A small crowd gathered along the walls of the garden to catch a glimpse of Albert and his fiancee.
Troops from the 87th Cadet Class formed a guard of honour and the band from the Defence Forces School of Music performed the national anthem of Monaco before the prince slowly stepped forward to arrange a ribbon on a large wreath which was laid in honour of the dead by two soldiers.
Drops of rain fell in the sunshine during the short but poignant wreath-laying and the minute's silence that followed.
The poem Rinneadh Aisling Duinn (We Saw A Vision), which is inscribed on the wall of the Garden of Remembrance, was read aloud in Irish by Lieutenant Tadhg Colbert of the 2nd Battalion, McKee Barracks.
As the Last Post was played, the Tricolour was raised to full mast.
The Irish national anthem, Amhran na bhFiann, was performed before the dignitaries left the Garden of Remembrance.
Albert kissed Ms Wittstock on both cheeks before he left in his car for a meeting with Taoiseach Enda Kenny at Government Buildings and his fiancee went to a private event at the Writer's Museum in Parnell Square.