Your cousin i.e. the employer needs to apply for an LMO with in Canada to HRDSC (unless your profession is already LMO exempt for some reason). In order to obtain an LMO, they need to have posted the vacancy for a good length of time (i guess it was 4 weeks) and demonstrate they were not able to fill it with any Canadians. Once he is able to obtain LMO (which would state your job nature, salary etc.), you can apply for a work permit in your home country using your job offer, LMO and your credentials related to job.
Current time lines for any of those are highly variable due to unanticipated situation in Canada (PAFSCO strikes, LMO rules tightened etc.).
But generally getting LMO would take any where between 6 weeks to many months. Then applying for visa depends upon your home country. Check average processing times for CIC for your home country or the nearest embassy which officially caters your home country. From inside Canada this can be as fast as 1 month if applied on line. From the rest of the world it could be as long as even up to a year for some countries.