If you enjoy walking and are not under pressure of time, you can happily and safely walk from Eason's in O'Connell St to Mount St/Merrion Sq area.
Cross O'Connell bridge, head towards Trinity College and Grafton St.
Turn onto Nassau St by the side of Trinity and thence to your apartment in Lr. Mount St / Merrion Sq.
Have a look at this city centre map for guidance. Your apartment is in the Quadrant D3.
You will be able to take advantage of visiting/seeing art galleries, museums and government buildings.
On Sundays in Merrion Square, there's an Open Air Art Gallery:
I'll leave the bus aspect of your question for others more qualified. When you are heading from O'Connell St to Lr. Mount St, you will catch the bus on the opposite side of the road to Eason's.
You need to be on the side of the road where Clery's - with its famous landmark clock - is located.
Enjoy your work and your visit to Ireland.