Unemployment insurance after salary reduced

I am working in Japan as an engineer since last 2 years. My company is paying unemployment insurance (shitsugyou hoken) regularly. My contract will finish soon and my company cannot extend my contract due to some funding issue. Since I have a visa valid until Jan 2020, so my company has told me that I can work as a part-time employee if I wish to continue with 50-60% salary of my present salary. I have also agreed to continue as a part-time employee and search a new full-time job in other place.

In this situation, can I apply for unemployment benefit?

If anybody has knowledge about this, please kindly share it.

No you can not.
Shitsugyo hoken is for people who is temporarily unemployed and looking for jobs. Being employed for low salary is not "unemployed".

If you have visa until 2020 that does not mean you have right to live until 2020. In your situation either live with salary you are getting or leave company and you have 2 months to find new job
If you cannot find in that time you need to go back to your country, that's how it works