The Ministry of Foreign Affairs website was updated on September 30th, and we can see the whole picture.

The home page is here https://www.mofa.go.jp/ca/fna/page22e_000921.html.
(It is a little unclear how to realize the description that "the number of immigrants is limited to a limited range" ...)

According to this, for medium- to long-term stay purposes, the following documents must be submitted to the Japanese embassy to apply for a visa.

１、Required documents at the time of visa application

(b) Mid to Long term stay

Based on the decision on September 25, 2020, all types of the status of residence, provided the applicant has the Certificate of Eligibility, are applicable for the new entry to Japan (Those who are entering Japan for “Diplomatic” or “Official” purpose do not need to follow the procedures for “Residence Track”) [Note 2]
(i) Visa application form (with photo attached)
(ii) Passport
(iii) Certificate of Eligibility [Note 3]
(iv) Written pledge (PDF)Open a New Window (2 copies) (Residence Track) [Note 1]

Note 1 and Note 2 are also extremely important.
Quoted below.

[Note 1] Please submit two copies of the pledge and the schedule of activities in Japan (“Business Track” only) at the time of submitting application (such as the original scanned and printed electronic data in PDF format). One copy will be returned, so please submit it to the quarantine when you enter Japan. Please keep the original for 6 weeks after the host company / organization enters Japan, and submit it when requested by the relevant ministries and agencies.

[Note 2] Specifically; “Professor”, “Artist”, “Religious Activities”, “Journalist”, “Highly-Skilled Professional”, “Business Manager”, “Legal/ Accounting Services”, “Medical Services”, “Researcher”, “Instructor”, “Engineer/ Specialist in Humanities/ International Services”, “Intra-company Transferee”, “Nursing Care”, “Entertainer”, “Skilled Labor”, “Specified Skilled Worker”, “Technical Intern Training”, “Cultural Activities”, “Student”, “Trainee”, “Dependent”, “Designated Activities”, “Long Term Resident” are applicable. Those who are considered as “Spouse or Child of Japanese National” or “Spouse or Child of Permanent Resident” don’t need to present the written pledge for application for visa, provided that they present their Certificate of Eligibility or full copy of the family register, etc.

If one submits these documents, those who have a Certificate of Eligibility will be granted a visa and will be able to enter Japan newly.

The most important of the required documents is the Written pledge signed by the receiving organization 
The link is here. https://www.mofa.go.jp/mofaj/files/100076147.pdf

Now let's dig deeper into the content of the pledge.

２、Important points in the pledge

First of all, this pledge should be written by the host organization in Japan.
If you are an office worker, your company has to write, and if you are a university student, the University has to write it.

It would be nice for a company, but it seems that it will take a long time for the university to make this pledge.

List the important points in the contents.

2 Protection Matters (1) when the subject is immigration from the target area of the refusal of entry
b subjects, subjected to the PCR test for the new coronavirus within local departure before 72 hours, from local medical institutions by using a predetermined format, Obtain a test certificate proving that it is "negative", and present and submit the certificate or a copy thereof to the quarantine officer and immigration inspector when entering Japan. In addition, the subject should understand that if he / she cannot submit the proof or a copy thereof to the immigration inspector, he / she will be subject to immigration denial based on the provisions of the Immigration Control and Refugee Recognition Act.

This is the infamous pre-immigration PCR negative certificate. 
The important point is that it must be in this prescribed format.

the prescribed format is here.https://www.mofa.go.jp/mofaj/files/100078393.docx

It became a hot topic on SNS etc., but there was a case where entry was refused just by writing "not detected" instead of "Negative" like a sample.
One will need to use this prescribed format and make a doctor write it as per the sample.

If one does not receive the PCR test within 72 hours before departure, one will not be able to enter Japan.

This is one of the more difficult conditions in some countries, as it is not possible in the medical condition of the home country.

2 Protection Matters (1) when the subject is immigration from the target area of the refusal of entry
c the target person joins private medical insurance (including travel insurance that covers medical expenses during the period of stay) at the time of entry.

Be sure to get travel insurance.
Since it says private medical insurance, it means that one can't select the health insurance getting after you come to Japan.

It's a very detailed point, and one will be granted a visa even if one doesn't have an private medical insurance, so don't miss it.

If one cannot show the insurance policy at the time of immigration, one may not be able to enter Japan and  may even be deported.

2 Epidemic prevention matters (1) When the target person enters the country from the target area of ​​entry refusal
m the host company / organization shall save the above (1) location information of the target person and the above (1) investigation. When one will be asked to provide necessary information as cooperation with, please obtain the consent of the target person in writing in advance to present the location information.

An "agreement letter" is required even though it is not written anywhere as a submission.

This agreement cannot be used in English and must be in Japanese or in both Japanese and English.

If one does not issue this agreement letter as a paper, when one will be asked to submit it at the time of immigration, and if not, one may not be able to enter Japan.
Make sure prepare the agreement letter, send the data to the foreigner in home country, let he/she print out and sign on, and then let them be with it.

(Information updated on October 1st)
When I checked with the Immigration Haneda Airport Branch, the officer said that one would not be asked to show this "agreement letter" at the time of immigration.
Therefore, it is not necessary to have it at the time of entry, but the agreement letter with the accepting institution must be kept by the accepting institution.

３、conclusion

From October 1st, in principle, all foreigners with a Certificate of Eligibility will be able to enter Japan.

If one has all the necessary documents, one should be able to pass the visa examination without any problems.

However, the immigration is not so.
Depending on the immigration officer, if he/she is a strict person, he/she may be asked to check the contents of the pledge.

One should take great care to bring them to Japan so that this does not happen.

In the worst case, due to incomplete documents, it is possible that one will be returned to a home country.

Below is a summary of the documents really needed.

Documents required to apply for a visa.

(i) Visa application form (with photo attached)
(ii) Passport
(iii) Certificate of Eligibility
(iv) Written pledge  (2 copies)

In addition, what is required for immigration,

・ Pledge
・ PCR test certificate within 72 hours before departure in the prescribed format (let's write it according to the sample)
・ Private medical insurance certificate (travel insurance is also acceptable)
・ Agreement letter with the host organization for cooperation about providing location information (Japanese)
and a copy of the Certificate of Eligibility

I think that one will be able to pass a normal immigration, but if an immigration officer is very strange person, one should call his/her boss and explain the situation.

In that case, it is also important to prepare a system that allows one to make roaming calls or Wifi calls with a mobile phone. It's too late after something happens.
I would like the all foreigners to be fully prepared and go through the immigration.

The transportation after entering Japan are limited to the following.

『Private car, host company or/ group owned car, rental car, hire』

Therefore, in the case of a company, it is better for the staff of the receiving organization to pick he/she up at the airport by car.

University students cannot be picked up by university staff, and they do not have a Japanese driver's license, so they need to reserve a friend's car or hire (taxi is not allowed).
Pay the latest attention even after entering Japan.

Since new entry has been eased in principle, please prepare properly and read the materials, so that one will not be repelled at the airport.

I sincerely hope that the foreigners will be able to enter Japan safely with a smooth visa grant and immigration.