A permanent resident can sponsor your common-law partner for permanent residence - but from your other thread, I'm confused as to whether or not one of you is a permanent resident. If so, the rest of this is applicable. In the meantime, I'll go log in on the other thread, too.
Not having valid temporary status in Canada does not make a partner ineligible to be approved - but if the restoration of status was refused, the refusal probably came with a demand to leave Canada. Applying to sponsor - even inland - does not cancel that demand, and not complying means you risk messing up PR eligibility. Some people will tell you that you should apply inland to avoid deportation - and to some extent, that may be true. But it will also put the both of you in a situation where the application has to go for special processing and it could be years before even valid temporary status is restored - let alone permanent status. In the meantime, only one of you will be eligible to work to support your family. Think very carefully before you back yourself into that corner. People who are not in Canada legally, who apply for PR via the inland process, do NOT get open work permits within 6-7 months after submitting their applications. That timeline ONLY applies to applicants who apply inland while they still have valid temporary status. It's too bad - if you've been common-law for three years - you did not apply to sponsor while the temporary status was still intact. Had you done that, you could have applied inland, with an extension application included, and that would have continued the work permit and protected the temporary status until PR was approved within normal inland timelines. Now, applying inland will be long and risky. If the applicant ends up having to leave Canada, even just for financial reasons, the application is forfeited.
It is possible to apply to sponsor via the outland PR process, even while in Canada. It won't keep an applicant who doesn't have valid temporary status from having to leave Canada - but having to leave will not affect the processing of the outland ap. The one thing you do need to be aware of with an outland ap is that any interview to verify the genuine relationship or the common-law qualification will happen at the overseas visa office - so to minimize the chance of an interview, make sure the evidences you provide with the application are quality. I don't know which country we're talking so I can't say how long processing will take, but I would suggest that you start immediately to minimize the amount of time you might have to spend apart.