You may not need to change the date with the UKVI when you wish to postpone your arrival for only a month. Under the Tier 2 general visa scheme, you would want your employer to agree to the delay. You're able to begin your UK stay up to 14 days before the start date on your certificate of sponsorship. That start date should help you decide whether any adjustment is even necessary.
However, should it be a major change of plans, ask the employer to update the certificate of sponsorship. It's an electronic record that has your details and those of the job. Your sponsor would have given you the reference number to include in your application.
The certificate number can be used only once, within 3 months of getting it. So, a major change and a new certificate number could alter your plans significantly, and may not be needed, given what you've indicated here.