once you are married, you will be able to sponsor him for PR under spousal sponsorship. if he wants to stay in canada while his application processes, he MUST apply to change his status to visitor and extend his stay. Not doing so will mean he is out of status, and that will mean he is in canada illegally and is subject to deportation. PR applicants are rarely denied extensions to stay, so there's no excuse for NOT applying to stay in status while applying for pr.
Right now, the best way to apply for spousal sponsorship is through the "outside of canada", or outland, option. a person can be IN canada as a worker, student or visitor when they submit an outland application. there is no rule against this. For most visa offices, outland is taking about 8-10 months for total processing, some are less (4-6 months) and some are more (12 months). The visa office is determined from where the applicant is from.
If you find out about applying "in canada", or inland, you will notice there is a new program that issues open work permits to inland applicants, and CIC is stating it will happen within 4 months. Since this is a new program, it is still unknown if the "4 months" starts from when the application is received or when CIC sends an AOR (acknowledgement of receipt). These are 2 different timelines, and noone will really know when the "4 months" starts until march. Please don't let this new program fool you that this will be the best option. for most people, it is NOT, and the OWP doesn't give much jump start in life in canada than applying outland. Currently, the inland process is taking 2 years to complete.
Also, don't let the processing times for visa offices on the CIC site fool you. What is listed are worst case scenarios and do not represent what applicants are seeing today. to get the best real time account of the visa office from where the applicant is from, you will want to take a look at the thread on this forum specific to that office. most threads have a spreadsheet that tracks progress where trends can be easily identified.
at this point, unless you are considered "commonlaw", which means you can prove you have lived together for 365 consecutive days, you can't do anything about sponsorship until you are married.