With the first category "Long Term Resident", its normally granted to those who have lived in Japan long term under a different category, such as Spouse of Japanese national and have either divorced or have lost a spouse through death and want to stay in Japan to maintain contact with children and or care for them.
The second category is as you have explained for (Long Stay for sightseeing and recreation)
However the period of stay would be for 6 months, which means to stay longer you would need to apply to extend before the initial six months expires.
You would need to provide proof of savings such as a Bank Book of more than 30 million JPY (At today's rates that would be above US$267,000).
You do not need to provide a Certificate of Eligibility, but by not doing so you would need to provide further documents, which would include 1) Schedule of your stay. 2) Documents such as a Bank Book to prove you have savings above 30 million JPY. 3) Documents to prove that the applicant holds private travel insurance which covers death, injury, illness during stay.
4) (if you apply from outside of his/her own country) documents to prove that the applicant legally resides or works in the country where the application is made
Specified visa: Designated activities (Long Stay for sightseeing and recreation) | Ministry of Foreign Affairs of Japan
With the COE you would need a Guarantor residing in Japan.
Supporting documents for the Long Term Resident Status COE application are:-
1) A copy of the family registration, the marriage certificate, the birth certificate and other documents certifying the status of the foreign national concerned.
2) Documents certifying that the person concerned can defray all the expenses incurred during the stay in Japan. In cases where the expenses incurred during the stay in Japan is defrayed by another person, documents certifying the income of the person who intends to defray all the expenses.
3) A letter of guarantee by the person living in Japan.