When did the astrologer usually starts his day’s business?

The astrologer usually begins his day's business at noon or midday, as the story goes.
According to the text, the astrologer is always punctual; this means that, at exactly the same time each day, customers can expect to find him sitting underneath the huge tamarind tree at Town Hall Park. There, he will have his paraphernalia spread out before him: a square piece of cloth with mystic charts on it, a notebook, a stack of palmyra writing, and a dozen cowrie shells.
The astrologer will be wearing his usual saffron-colored turban, and his forehead will be smeared with sacred ash and vermilion. The text tells us that large groups of people will pass by the astrologer's station throughout the day. Perhaps, the astrologer begins his work at midday because that's when the crowds begin to swell.
From the story, we learn that the astrologer is efficient in how he conducts his business. Because of his experience, he can decipher each customer's concerns within five minutes of their initial meeting. Then, after ten minutes of his customer speaking, the astrologer will be prepared with his diagnosis and answers to anxious questions. He is always careful to present a respectful attitude, regardless of each customer's foibles and predilections. The astrologer is a clever businessman and usually earns enough money for his daily upkeep by nightfall each day.

"An Astrologer's Day," the titular story of a collection R. K. Narayan published in 1947, begins with an explanation of the astrologer's daily schedule. This is the first line of the story:

Punctually at midday he opened his bag and spread out his professional equipment. . . .

So the answer to your question is "midday," or noon. It seems a little late for a person to be starting his daily work, doesn't it? But Narayan continues to offer details of the astrologer's schedule and the circumstances of his professional life throughout the story. For instance, he describes the crowds that surround the astrologer. He sits, with his tools spread out before him, amid other vendors on a street that runs through a large park:

A surging crowd was always moving up and down this narrow road morning till night. . . .

The crowd will be there through the night, Narayan writes, and we can assume that's why the astrologer doesn't start working until noon.
Also, the story reveals, the astrologer's schedule depends on having enough light to see and to do business. The area where the astrologer works is not illuminated by municipal lighting, so the vendors are partly dependent on each other for light sources:

The nuts vendor blew out his flare and rose to go home. This was a signal for the astrologer to bundle up too, since it left him in darkness except for a little shaft of green light which strayed in from somewhere and touched the ground before him.