"I walk with my cat on Sundays."
Translation:Siúlaim le mo chat ar an Domhnach.
"On Sundays" here means habitually, so you would use Ar an Domhnach. You would never use domhaigh with the ar an structure. However, if you are talking about a particular Sunday (usually the one coming, ex "I will walk my dog Sunday") and not habitually, you would use Dé Domhnaigh.
Siúilfaidh mé mo mhadra Dé Domhnaigh (I will walk my dog Sunday)