If it has the form Dé X or ar an X, then it’s an adverbial form, and in English the adverbial form can be either “on X”, “on Xs”, “X”, or “Xs”, e.g. “I’ll see her on Monday”, “I see her on Mondays”, “I saw her Monday”, or “I saw her Mondays”. The Dé X form will typically not translate as one of the “Xs” forms. The X in Dé X is genitive, which is why e.g. it’s Luain rather than Luan in this exercise.
An Luan without a leading ar is the noun “Monday” rather than an adverb, e.g. Is fuath liom an Luan (“I hate Monday”).
"Ar luain" is a poor attempt at a literal translation, it's both ungrammatical and inaccurate.
Luain is the genitive of Luan, so it is used after the noun Dé, but not after the preposition ar. When used with the preposition ar, Luan will generally require a definite article - ar an Luan.
"I read the Sports section on Monday" - Léim an roinn spóirt ar an Luan
"I'll see you on Monday" - Feicfidh mé thú Dé Luain