Shinjuku Gyoen National Garden
Trees, lawns and a change of pace near Shinjuku. A paid garden visit with seasonal hours, worth giving a little breathing room.
Notes from the cityTokyo
A free building, a paid exhibition, a collection ticket and a timed visit are different things. Start with the experience you want.

Kaz Ish / Wikimedia Commons. Captured: Taken on 16 February 2013. Resized and compressed; display crops vary. · CC BY-SA 3.0: License (including resized version where share-alike applies); Source
The right question is not just whether a museum is free. It is which part of the visit you want, on which date, and what that ticket includes.
At the National Art Center, you can enter the building for free, but exhibitions have separate admission. At Tokyo National Museum, a special-exhibition ticket also includes the collection on the same day. Mori Art Museum and the observation deck in its tower are different experiences; read the ticket description rather than assuming they come together.
Dates matter just as much. Holiday openings can move a normal closed day, and an exhibition can have its own late sessions. Our event pages use the published visiting intervals. Start there, then follow the official source to confirm the ticket or any last-minute notice before travelling.
Trees, lawns and a change of pace near Shinjuku. A paid garden visit with seasonal hours, worth giving a little breathing room.
Give Ueno a museum day. Start with the Japanese galleries, then choose a smaller corner of the collection to linger in.
A free temple visit in Asakusa, with the great gates and main hall as an anchor for exploring the neighbourhood.
Contemporary art on the 53rd floor of Roppongi Hills Mori Tower. Choose the exhibition before choosing your date.
A glass-fronted art destination in Roppongi. The building is free to enter; exhibitions have their own tickets.