The additive noise power is the average power of Gaussian noise. It could be estimated based on the noise figure of the receiver. The noise power=K T_0 F B. K is Boltzmann constant = 1.38*10^(-23), T_0 is the equivalent noise temperature and it is usually taken as 300, and F is the noise figure of the receiver and B is the system bandwidth. For some practical systems, F can be around 2-8 dB (can be more in some systems). Remember to convert from dB to linear scale before substitute in the equation. If you use normalization (i.e., consider unity channel gain, then remember to divide also the noise power).  

If you know the SNR before beamforming then you can calculate it after by noting that you get a coherent integration gain of M*N, where M is the number of elements in your antenna array and N is the number of snapshots (in time). This is because you are performing coherent integration (akin to FFT) both when you're summing voltages across elements and summing the (I presume) signal covariance matrix in time.

If your output is a linear combination of the values before the beamformer, then your noise is a linear combination of gaussian random variables, being also gaussian. Its variance may be calculated by using the same linear combination of input variances but with square coefficientes (e.g., output = a input1 + b input2 => var(output) =a^2 var(input1) + b^2 var(input2).

Notice that this property is only valid for gaussian variables, and only if your beamformer is linear.