Is it possible to supply Arduino nano with LED-stripe from 12V
I remember Answering similar question somewhere. Do not worry about Arduino sharing 12V supply with LEDs. Any supply with CE mark (I would have expected that to be standard in Russia and nearby countries) would be unlikely to spike above the means of arduinos voltage regulator.
There is a small chance that your supply may assume the consumption is too low and go to sleep, but that is rare.

I guess the question is a matter of controlling the LED strip. from the strong 40Amp I am guessing that LED strip would be longer then few CM, So transistor wont cut it 
Its a question of MOSFET vs relay. MOSFETS are better if you keep switching LED's on and off a lot, few Milion times is no problem for them, however they are expensive especial if there is a high current (the 1K resistor is not necessary, just a habit to protect the output).







simulate this circuit &ndash; Schematic created using CircuitLab


Second option is a relay. its useful even on a higher load, but only gives several thousand, and in case of e-bay special like songle about a hundred switch of's(or ons). Note that many relays have two ratings in Amperes. The smaller number is for you here (DC).







simulate this circuit

Here 1K is required

Although when saying 40 ampere dont forget the rule of online shoping on power )supply, light emission etc: 
50 Ebay Amperes = 60 BangGood Amperes= 80 Wish Amperes= Around 18 Real world amperes

So if your 40Amp supply is from ebay that would make it about 16Amp of long-term (more then half an hour-10 hours) running time