My favorite musician, Ryuichi Sakamoto

He was a member of YMO, a popular music group in the 1980s.

They were cool guys among young people at that time.

This isn't wrong, but it sounds kind of awkward in English. Try something like: "The youth of the time thought they were very cool." Your sentence sort of implies that they were among the young and stood out from them because they were cooler than others. I imagine what you're trying to say is that young people thought they were very cool.

Time has passed and young men have become old men.

The passage of time allows us to pass from the past into the future.

Ryuichi Sakamoto is 60 years old now.

Years is not possessive or a contraction, so no apostrophe.

Although he has lost (or is losing) his good looks, his musical talent never gives out.

"Loses" makes it sound like it is something that happens every so often. Once looks are lost, they're lost--you can't lose them again. If he still looks sort-of good, then use the progressive "is losing."

The following music clip is from the recent YMO's performance.

The clip is only a part of the performance (that's what makes it a clip). When you say "the clip is the performance," you sort of imply that it is the performance in the entirety. A clip, by its nature, is only part of the whole. If you were linking the whole performance, I would use "video" to describe it.

Recently, he often attends the anti-nuclear demonstrations.

Since you use "often," you are not talking about a specific anti-nuclear demonstration, so the singular "the anti-nuclear demonstration" is incorrect. If he only attended one, and everyone knew which demonstration he attended, you could say "the anti-nuclear demonstration."

The other day, he came back to Japan from New York and attended the anti-nuclear demonstration in front of the prime minister's official residence.

Time to be super pedantic! "Demo" _is_ short for "demonstration," but a "demo" is almost always used to refer to a demonstration of a product for purposes of evaluation. I would hesitate to shorten "a political demonstration" to "a political demo."

Title are only capitalized when they are being used to refer to a specific person, or when the precede a name: "Prime Minister Yoshihiko Noda" or "Yesterday, I saw the Prime Minister." When not referring to a specific individual, it would be lower-case. "There have been 3 different prime ministers in the last several years." Because the Prime Minister (Noda-san) lives in an official residence where many other prime ministers have lived, we would probably say "the prime minister's (position, not referring directly to Noda) official residence."

"Official Residence" would never be capitalized, unless it was a name like "the White House" or "Gracie Mansion" (the official residence of the mayor of New York).

I respect his actions.

However, I didn't think that he was the type of man who would attend such a demonstration.

Almost all public entertainers keep silent about nuclear power plants in Japan.

Because They know that if they attend anti-nuclear demonstrations, they will have their livelihoods taken away.

Many entertainers have many livelihoods. You could merge this with the last sentence and keep the because if you like: "Almost all public entertainers keep silent about nuclear power plants in Japan, because they know that if they attend anti-nuclear demonstrations, they will have their livelihoods taken away.