How would I analyze the meaning of John Lennon's song "Borrowed Time" as a music critic? Any suggestion would be very helpful and appreciative. I need to stay focused on the argument by asking how is the music organized, what unifies the music, what style is the music, and from what period. What is the larger historical context in which the song was created?

You should analyze song lyrics in the same way that you would analyze a poem. As for the music itself, you need to identify the genre, which is pop. The period is simply when it was created. This comes from Lennon's later career, when he worked solo and with his wife, Yoko Ono. He recorded it in 1980, shortly before he was assassinated, but it was released posthumously in 1984. So the period of the song is the early 1980s, and like many songs of its time, it has a synthesized pop, new-wave sound, with a little bit of psychedelic funk thrown in. 
The lyrics themselves are pretty straightforward, so you should first identify the main theme, which is about growing older, and how aging has brought clarity and greater happiness to the writer. Lennon equates being younger with psychological pain, confusion and delusion.
The reason that the writer is happier now that he is older is that he has come to realize something very basic but important. This realization comprises his chorus, which goes like this:

Living on borrowed timeWithout a thought for tomorrowLiving on borrowed timeWithout a thought for tomorrow

This is just a simple "seize the day"-themed poem put to music, and while it is perhaps overly simplistic, it gained a great deal of poignancy when Lennon was shot and killed a few months later just outside of his apartment building by a deranged fan, Mark David Chapman. It is interesting to see how the circumstances of Lennon's life and tragic, violent death at such a young age imbued what is essentially an upbeat, fluffy, "living for the moment" song into a sadly prophetic exhortation about a man who is happy to be alive and is embracing each moment, when as we the listeners now know, he had so little time left on earth.
This song is a good case study for how circumstances surrounding a piece of art can transform that art into something far more powerful and meaningful than the artist intended or would have achieved otherwise.