Can I register an SSL certificate that will only start to be valid in the near future and not right away?
Ordinarily, the validity of period of the certificate is not determined by the requestor, and is not specified in the CSR. The validity period is determined by the CA (most CA's offer various durations according to price). So, you would have to find a CA that has a mechanism in place for what you are trying to do.
See CSR expiry date / validity date for more info.