The safer way is you cut the vector pET-51b with NcoI and any other appropriate enzyme (downstream of NcoI site, let's say HindIII), gel-purify it and then ligate it with your target gene (pre-cut with NcoI + HindIII) by using T4 DNA ligase, and then do transformation with any E. coli cloning competent cells and screening.

For the initiation of translation, the start codon should not be too far downstream. The start codons of most expression vectors are usually in the optimum distance from the ribosome binding site. Removing that codon and having one further downstream of that could affect the efficiency of the initiation thus the yield of the protein. I don't know if there is a clear maximum for the distance where the start codon should be, but I would say it's more a gradual decrease of efficiency.

Likewise, if you have a second ATG codon in a short distance downstream of the start codon, you could get two versions of your protein with different N-termini.

I would agree with Hüseyin Besir that the problem is that pET plasmids are optimized in terms of the ribosome binding site and its position relative to the plasmid ATG start codon. So if you use the start codon of your gene you need to be sure that your insert includes the native ribosome binding site (assuming it is a bacterial gene). If not, then it is likely your expression will be low. Changing the distance between RBS and ATG by even a few nucleotides is enough to significantly reduce efficiency. The best strategy is to amplify your gene by PCR and introduce the NcoI site at the ATG start codon so that you can clone directly into he NcoI site of the vector

I agree that the best option is to remove the tag rather than the START codon. If you have already cloned your GOI into the vector, the easiest option would be to run PCR with primers running outside of the tag with some appropriate restriction enzyme (ideally NcoI as it contains the START codon so you minimize number of extra AA residues), cut and self-ligate.