I would say no.
A plot is a piece of land for building a house on, or a single allotment in an allotment garden, but I wouldn't use it for an agricultural space that is worked by a farmer.
I would most certainly use 'plot' as an agricultural space for a farmer, but generally it does imply a smaller area of land than a 'field'. For example, a subsistence farmer in a developing country would have several plots of land to grow crops.
In my usage of native English English a plot is a area of land which may or may not be used to grow crops or graze animals, it could have buildings on it. In fact, the phrase a "building plot" is in common usage, meaning an area of land designated to be built upon. Except in very special cases a field is an area of agricultural land with boundaries.
"the farmer has lots of fields" is currently marked as incorrect. "A lot", "lots" and "many" are all quite interchangeable, I'd say, although the first two are a bit less formal than "many".
If it matters, "lots" is not accepted :)
Mine: the farmer has lots of fields
Shown: The farmer has a lot of fields.