I was walking down the corridor and noticed that one of the bulbs there has burned out

[EngLearner]

Peter is John's boss. To get to the room where they work, he has to pass through a corridor. While passing through it, he notices that one of the bulbs there has burned out. After entering the room and taking off his coat, Peter addresses John:

John, I was walking down the corridor and noticed that one of the bulbs there has burned out. Could you please have the janitor replace it?

Are the tenses in bold used correctly in the example that I made up? "Has burned" doesn't need to be backshifted to "had burned" in this case, right?

 

[emsr2d2]

Peter is John's boss. To get to the room where they work, he has to pass through a corridor. While passing through it, he notices that one of the bulbs there has burned out. After entering the room and taking off his coat, Peter addresses says to John:

John, I was walking down the corridor and noticed that one of the bulbs there has burned out. Could you please have the janitor replace it?

Are the tenses in bold used correctly in the example that I made up? "Has burned" doesn't need to be backshifted to "had burned" in this case, right?

You're right - no backshifting is required there. If he were relating the story to someone later, he would say "When I was walking down the corridor, I noticed that one of the lightbulbs had blown".
(In BrE, we tend to say that a lightbulb has blown/has gone". Note that "gone" does not mean "vanished" there.)
You didn't need "there" either time. It's clear from context where he's talking about.
Also note that I would say "has to walk down/along a corridor", not "pass through".
"Addresses" is a bit formal for this context.

A shorter and, in my opinion, more natural version would be "John, I just noticed that one of the bulbs in the corridor has blown. Can you get the janitor to replace it please?"