You may think you need to use loop equations to solve the problem, but if you look closely at the second loop you will see that the current through R1 is defined by the 2mA current source. Therefore, the voltage across R1 can be found simply by Ohms law: V_{R1} = I_{R1} × R1 = 2mA × 1kΩ = 2V.

Sure, if you wanted to calculate all the node voltages with respect to ground then you would need to use either loop equations or network reduction, but you don’t need that to answer the question as stated.

The moral of the exercise is to always look for the simplest solution first, and don’t make the problem harder than it really is.

I was tempted by this response of considering only the current driven by the current generator at the beginning but was it really correct?

I mean it seems to depend on the model you choose for generators and can you be sure that none other branches will add a contribution?

Applying Kirchhoff's circuits laws seemed not to be realistic in a decent time...

So I used the superposition principle: as soon as you consider another generator than the current generator close to R1, this one can be replaced by an open circuit (ideal Norton model) which demonstrates that no other generator has any contribution to the current through R1. Q.E.D

And if you want to use a bit more realistic model (very high resistance instead of open circuit) the superposition principle still applies and you end up with a sum of 6 terms for defining the current through R1 instead of ... many equations to solve!

Thanks for this little exercise, back to basics!