Definitely yes. The primers have to be designed to anneal with the gene sequence you are looking for. You need not digest the plasmid in most cases. Make sure to run a gel with the plasmid and the amplified DNA later to verify if there would be any dimers or unspecific PCR product/s. Most of the primer design software (e.g. Vector NTI) can give you the right Tm, but you may have to do a temperature gradient if you want a very good product especially if you will use the gene for qPCR or transcription studies.
There is no doubt why the PCR will not work....during cloning in most of the labs...inserts are analyzed by colony PCR on regular basis.....all you need is a specific set of primer.....as suggested above by Dinesh..you can use Vector NTI for designing or use primer BLAST at NCBI...for designing specific primers...