It is possible to charge the Monitor One with a solar panel. The video in this post will be helpful. It's for the Tracker One, but they have the same Tracker SoM inside and will work similarly.
You can use a bare 12V panel connected via the M12 8-pin connector; you don't need a separate charge controller.
The amount of time it will run will be dependent on the size of the panel, which sleep mode you are using, and how often you publish. The first step is to use the battery calculator to figure out how much power you need. Then adjust that for the size of your solar panel and how much cloudiness you need to design for.