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When we see Varys writting letters for the second time in season 8 episode 5, he is picked up by Greyworm.

Varys hears them coming, half heartedly burns the letter and takes off some jewelry, where we get to see a thin ring in particular.

It quickly becomes apparent that Varys knows he will not live to see another sunrise and lives his last moments as planned and calm as if he had written the script himself.

I can see him burning the letter in order to not give up the recipient in the opening, but what about that ring? Why is it important enough for him to take it off? Why is it important enough to show us? Why does he prepare his last walk by taking off that ring?

There was nothing special about the ring. The action, however, served two purposes.

In Universe it is a ritualistic thing. Varys has always seen three steps ahead and exudes an air of complete control. He knew the likely outcome of his actions, and he deemed them to be worth it.

Varys: You know where my loyalty stands. You know I will never betray the realm.

Tyrion: What is the realm? A vast continent, home to millions of people, most of whom don't care who sits on the Iron Throne.

Varys: Millions of people, many of whom will die if the wrong person sits on that throne. We don't know their names, but they're just as real as you and I. They deserve to live. They deserve food for their children. I will act in their interest, no matter the personal cost.
_{Game of Thrones - S08E04: Last of the Starks}

When he hears the footsteps, he knows whats happening. As you touched on in your answer, "It quickly becomes apparent that Varys knows he will not live to see another sunrise and lives his last moments as planned and calm as if he had written the script himself." He has accepted his fate and so he prepares himself for death.

Out of Universe, it serves to give the viewer a glimpse inside Varys head. For us to know that Varys understand whats about happen, he has to take action. Think of the knock on the door earlier in the episode, we were worried, but he did nothing but hide the letter and look up. Based on our relationship with Varys, we knew he was still under control of the situation. This time, we all hear the boots, and then Varys burns the letter and starts removing rings. We know he is about to die, and now we know that he knows he is about to die.

So its mostly a cinematographic trick to give us, the audience, a window into the world and minds created by the show.

Varys has few possessions beyond his clothes and his rings. He also still has at least one girl loyal to him, an heir to his belongings perhaps.

Given that he knows Dany is fond of incinerating her enemies and his rings would likely melt, he takes them off to preserve the only things he seems to own, perhaps to pass them on or perhaps just so that there is something of him left in the world.