If you are looking for a quiet place to view the ume, or plum blossoms, and you are visiting Yamanashi prefecture, Furouen in Sakaori would be a fine choice. It is located only a 5 minute walk from the closest JR station of Sakaori. Sakaori Station is just 5 minutes away by train from Kofu Station, one of the major stations in Yamanashi.
This plum blossom Park is located in a quiet corner of Sakaori, Yamanashi. Not many people know this spot for plum blossom viewing because it is overshadowed by Maizuru Park in the heart of the city. This year viewing has been open since the middle of February, and the park will be open until the end of March. It only costs 500 yen for adults and 200 yen for kids. At the ticket entrance you will get a map of the park. This place is not so small despite the looks from the entrance, but since it is on a hill, you have to climb a little and use a bit more power to reach the top. The entrance to the park is hidden in a residential area (near Yamanashi Gakuin High School) with few if any signs pointing the way, so at Sakaori Station ask the station officer for directions, or follow Google Maps.
The park reportedly has 20 different varieties of plum trees that bloom, with about 3000 trees total whose blossoms you can enjoy. The park is open from 9:00 a.m. until 5:00 p.m. Both sweets and amazake (sweet sake) are available at the Ume Matsuri festival.
They provide plenty of benches on the top of the hill and in other spots, so you can sit back and enjoy viewing the blossoms. I recommend the top, where you can sit on a bench and enjoy the view of the blossoms with Mount Fuji on the other side. Not only can Mount Fuji be seen on clear days, you can also see the peaks of the Japan Alps and other mountains near Mount Fuji.
After enjoying the view, climb down from the west side and visit the gift shops and food stalls along the way to enjoy some small snacks while chatting with your friends. Pets are also allowed in the park. If you want to take home an ume tree, there is even a little nursery that grows several kinds of the plums. You can buy them for a reasonable price, as souvenirs or to plant in your garden.
Official website of Furouen Park: http://www.furouen.jp/english