Invariance of $T[x[n]]=\sum_{k=n-1}^{n+2}x[k]$
Yes your computation is correct. Indeed, you could have seen this even easier, had you considered that the output computation sum was

$$ y[n] = T\{x[n]\} = x[n-1] + x[n] + x[n+1] + x[n+2] .$$

And this is clearly an LTI (linear time-invariant), FIR (finite impulse response) system with associated impulse response 

$$h[n] = \delta[n-1] + \delta[n] + \delta[n+1] + \delta[n+2] $$