What you are hearing is a "flapped" r sound (which does sound like a D) followed by AN, but the AN is reduced to A', so you get "ar a'". You sometimes hear a flapped R in "upper crust" British speech in, for example, "mirror" which sounds like [midda] as opposed to standard American, which sound like [meer] to my British ears.
No. It should be accepted, and, in fact seems to be the preferred translation for ar an Máirt (Meaning ar an is generally habitual, Dé ____ for specific one)
Ar an Máirt = On Tuesdays. Máirt in singular in the same way that Tuesday is singular in the dialectal English expression "I walk of a Tuesday".
"On Tuesdays" is the right interpretation because Irish present tenses are "habitual" in essence. If you want to be specific about a particular Tuesday, I suppose you could use the continuous, "Tá mé ag siúl Dé Mairt" or "Beidh mé ag siúl Dé Máirt".