Tower flux measurements using Eddy Covariance method would provide the amount of net CO2 uptake by ecosystem (plants and soil) during daytime and CO2 release by ecosystem during night.

These are pretty standard and well established measurements. Free PDF book is here: https://www.licor.com/env/products/eddy_covariance/ec_book.html

One can then model CO2 uptake by plants only using ecosystem-specific time-specific NPP model.

With oxygen it is more difficult because  instruments are not readily available, although they are getting better/faster. It may be possible that one can model oxygen release from carbon uptake if plan specie and growth stage are known, but I am not very familiar with this and cant comment.

At the field-scale, eddy covariance is your best bet. You could alternatively examine these processes at the leaf-level with the licor 6400 portable photosynthesis system. It measures a slew of parameters including CO2 flux . It does not measuring O2 flux, but does measure water flux. Importantly, these fluxes will vary according to all sorts of environmental conditions (weather, soil properties, fertilizer and irrigation regime), so if you are trying to find some sort of average value for a crop, it may take a great deal of data to do so.

It's based on photosynthesis (tree absorb CO2 and release O2) and respiration (opposite with photosynthesis) process of the tree/crop.

Depending on C3 or C4 trees, some documents show that respiration can reduce 30-40% productivity of photosynthesis.

By measurement the carbon biomass, we can calculate volume CO2 absorbed.

For larger scale, The ex-ante Carbon balance tool (that's developed by FAO).

Now days , CO2 probe is also available , ir can measure the rate of release of CO2 directly from the soil. That should not be so difficult 

The ability of photosynthesis to release Oxygen is based on the equation Carbon dioxide plus water give carbohydrate plus oxygen.

The amounts of Oxygen that are released are proportional to Carbon dioxide utilized for photosynthesis by the green plant.

Thus the amount of photosynthesis as determined will correspond with an equivalent amount of oxygen released.

Photosynthesis is a light mediated reaction and most of the Oxygen release will be in the day and most carbon dioxide release will be at night. 

In terms some sense the Photosynthesis minus Respiration is equal to either soil stasis, soil regeneration or soil degeneration. Soil degeneration occurs when respiration outstrips photosynthesis and soil regeneration occurs when photosynthesis outstrips respiration leaving a dividend of accumulated soil Carbon and Nitrogen which improves soil fertility.