Train - Your best bet would be to make your way from either Heuston or Connolly Station to St. Stephens Green and catch the Luas Tram (see below). From Connolly Station you can catch the DART to Grand Canal Dock Station but to be honest the Mespil Hotel is still a good 10 minute walk away along the canal.
Bus - The most convenient Dublin Bus to use is bus no. 10, which runs from Upper O'Connell Street in Dublin city centre. There is a bus stop about 50 metres away from the Mespil Hotel at Baggot Street Bridge, just head for Mespil Road and you'll find the hotel on your left.
LUAS Tram - Catch the Luas Tram from St. Stephens Green (green line) and get off at the Charlemont stop, which is two stops away. When you get off the Luas, head onto Grand Parade and then Mespil Road, following the canal. After about a 5-minute walk, you'll see the Mespil Hotel on your left.