"He runs for the day" is basically the same thing as "He runs for a day". The day is conceptually the same thing as "a day". The context can be today, tomorrow, or some undetermined in the future, but they both mean the same thing.
"he runs all day"???
More like "He habitually runs for the period of a day"
Why is day singular? I thought 'lae' was the plural.
laethanta is the plural of lá.
Lae is the genitive - an extremely literal translation would be "he runs for the duration of a day", where the genitive is more obvious. Compound prepositions like ar feadh trigger the genitive.
Is "He runs during the day" acceptable?
I thought "ar feadh" means "during". But it appears that you must use "i rith an lae" for "during the day". It seems that in this case "ar feadh" means "whole" or "entire".
That's what I thought it meant, but it didn't give it to me. "He runs for a day" is a strange sentence in English, so I thought I was making it more natural. I guess I was wrong.
Could "ar feadh lae" also be translated as "all day long"?
"all day long" is an lá ar fad - ar feadh is best understood as "for (the duration of)" when used with a time period.
Ní fhaca mé thú ar feadh na mblianta! - "I haven't seen you for years!"