The Shinjuku Gyoen National Garden is a gorgeous park that's just a short walk from Shinjuku Station. There are several gardens within the space, including a formal French one, an English landscape garden, and a traditional Japanese design. While the admission fee is nominal (about $2), it helps assure that it is surprisingly quiet, with fewer visitors than parks open to the public for free. If the weather is good, consider picking up a bento from nearby Takashimaya's depachika. Convenience stores sell plastic "blue sheets" for impromptu picnics. The only downside to this park is that it is alcohol-free; if you want to drink sake at your picnic, head down the road to Yoyogi Park.