I'm looking for a data that what is the typical application range for a common uni directional strain gauge like used in weight scales? I've read about around 200 ppm in a research gate article, can I measure 40 ppm with a cheap setup (arduino nano plc, industrial grade op amp, soldered connections)?
I think it will come down to the quality of your amplifier. Let's put some numbers together.
With 10 volt excitation (typical) and a gage factor of 2.2 (typical), and full bridge wiring as per line 9 in this reference (https://www.hbm.com/en/7163/wheatstone-bridge-circuit/, there are other ways to do a full bridge but this is common), 40e-6 in/in would give an output of 0.5 mV.
In other to make a good measurement, you'll want a minimum resolution on your analog to digital converter to at least 10 times less than this (i.e. if your minimum resolution is 4e-6, then you can distinguish 40e-6 from 36e-6 or 44e-6, but no finer. That's probably the minimum acceptable and might want something a lot better depending on your requirements). So let's stay you need to be able to read 0.05 mV (i.e. 50 microvolts) with your ADC. Averaging multiple values can help increase resolution, but only to a point. you need a decent enough resolution to start with.
Arduino nano has a 10 bit ADC with a 5V range (https://store.arduino.cc/usa/arduino-nano). So 1 least significant bit is 5 / 1024 ~= 5 mV.
Thus you can read 5 mV and you want to be able to read 0.05 mV. So your amplifier needs to be able to provide an amplification of 100:1 (without introducing any non-linearity, distortion, noise, etc). You'll also need to keep the noise on the input side of the op amp to on the order of 0.05 mV (or else the op-amp will just amplify the noise along with the signal).
A professional quality strain gage amplifier which would meet those specs (e.g. Omega DMD-466) is going to run \$400-\$500 or more. I doubt you'll be able to meet the same performance with a "cheap" setup, although I won't say for sure that you can't do it.