"On a Sunday" would be referring to the idea of the day. So, for example, "it's always best to put out the recycling on a Sunday". That sentence just refers to putting out the recycling on a Sunday. "On Sunday" would be referring to the idea of doing something regularly. So, for example, "I always go to the supermarket on Sundays". That sentence refers to doing an activity on a regular basis. You may not use "Riding the circular Yamanote Line on a Sunday in Tokyo" to talk about doing something on a regular basis. If you wanted to talk about doing something regularly on Sundays that involves this sentence, one might say "Yoko has a very regimented schedule during the week. A part of this schedule includes riding the circular Yamanote Line on Sundays in Tokyo to get to her mother's house." I hope that makes some sort of sense.

You can say, "Most Sundays, I take the train" or "On Sunday, I take the train". Personally saying 'on Sundays' would sound grammatically correct depending on the context. For example, "I work on Sundays" is a grammatically correct sentence, while "I take the train on Sundays" does not sound entirely correct.

Yes, if it's habitual, you can say "on Sundays". Normally, you wouldn't say "a Sunday". It's only if Sunday is an adjective or adverb I believe, and not a noun. For example, "On a Sunday evening (evening is the noun)..." However, like in the news article, in the sentence given, "on a Sunday" is a prepositional phrase acting also like an adverb, so saying "a Sunday" would be when Sunday acts as an adjective or adverb in a sentence. Hope I made sense!