If someone died in the workhouse and no-one claimed the body they would likely be buried in a common grave, often in the workhouse grounds. (There is a burial ground shown in an 1849 map of this workhouse). But not everyone was buried there. If Albert had any relatives nearby they may have buried him in a family plot.
Here’s a link to information on the workhouse (originally the South Dublin workhouse). Their records are reportedly on findmypast.ie Can’t say whether they include those within the past 100 years. Possibly not. If they still exist, you may be able to make a Freedom of Information request to view them (to the National Archives in Dublin). If approved you’d get a redacted copy.
http://www.workhouses.org.uk/DublinSouth/As Hallmark has mentioned, most workhouses had hospitals attached and in later years of the system many people admitted themselves there to get medical treatment, rather than because they were totally destitute. (There seems to have been some flexibility around the admission rules). The admission records usually show whether an inmate went straight to the infirmary, or whether they were already in the workhouse.
After the workhouse system was closed down, many of the hospital wings continued as conventional hospitals, and that seems to have happened in this case.
