Career

How are you ready for a job?

Well, In my case, I start looking for jobs by checking out job offers on websites.

If you have personal connection to get a job, it is better than websites because it is easy to understand the relationship among people who serve the job.

Basically, in Japan, people have been working for 8 hours a day, and the five-day work week is adopted.

Even on Friday, they work the same as other weekdays, unlike other countries which shorten work time on Friday.

I wish I knew which countries do this... it sounds nice!

To have difficulty finding a job is social problem in my country.

Maybe,

"The difficulty of finding a job is a social problem in my country."

Young people who graduated from a university strive to get a job desperately, but it is so crucial that even the people who apply for more than 20 companies cannot get a job surely.

This is a very large problem in America too. Students also have to pay back their university loans, but without jobs they have no money and are unable to pay them.

Unfortunately, the rate of suicide increases because of this.

Companies tend to hire the people who work part time and as a contract employee.

Yes! It is very unfortunate.

It is because they can deal with temporary staves more easily than regular employees thanks to less responsibility.

I'm not sure what you mean by "staves" here.

Careers

Your title is general so feels right to pluralize to talk about careers in general. Otherwise you could say, "My Career"

How ready are you for a job?

You sound like you're asking "what steps have you taken to prepare yourself for a career/job" but it sounds a bit odd, otherwise you could ask (as in my correction) What state of readiness are you in regards to attaining a job.

Well, in my case, I started looking for job by checking out job offers on websites.

past tense 'started'

If you have personal connection to get a job, it is better than websites because it is easy to understand relationships among people who work at that job.

this is a bit of a strange sentence. I'd probably say this:

Job hunting from websites is not as good as finding a job from a personal connection. This is because it's easier to understand and get along with coworkers who you already know.

I think human interaction is totally important for any jobs.

Human relations isn't wrong however it isn't used a lot in regular speech and instead is used mostly for a job in a company "human relations" or "HR"

In Japan, 8 hour days, 5 day work weeks has been adopted.

Even on Friday, we work the same amount of hours as other week days unlike some other countries which shorten work hours on Friday.

corrected to we (Japanese people yourself included?)
they implies you are outside of the group of people
you could also say "...workers in japan work the same..."

~ I wish they'd shorten hours on friday here!

Difficulty finding a job is social problem in my country.

Young people who have recently graduated from university desperately strive to get a job, but it is so hard to get a job that even the people who apply for more than 20 companies are not guaranteed to get one.

Unfortunately, the rate of suicide also increases because of this.

It is because they can deal with temporary staff more easily than regular employees thanks to less responsibility.

staff does not get that plural. Staff is used to indicate more than one worker usually, a group of people (as one thing)