Basically I am going to be in Galway by myself for 4 days in a couple weeks and one day will be a Sunday. I am curious what I should expect to be closed? I picture some things will be probably be closed all day.
Fortunately for you, There has been no recent legislation regarding Sunday trading in Ireland, which is regulated by the Shops (Hours of Trading) Act, 1938.. As the link indicates, the act is mostly inoperative, and:
"...as a result most shops and businesses may open whenever they please, including on Sundays and public holidays."
Supermarkets and convenience stores are often open longer on Sundays.
The only small issue if you want to get started early, is that off-license shops that sell alcohol are not permitted to do so on Sundays until after 12.30pm.
You will find that a lot of locally-owned, non 'big name' main street shops wouldn't open, but those in shopping centres (malls) will all be open, as will anything involved in the hospitality industry. You may find a few things don't open until midday though.