My wife will be relocated at the end of this month to another country and I will stay in HK. I'm on my wife's working visa as we are married and I'm working in HK. What are my options if I stay in HK?
Can I keep my working visa until the expiring date?
Do I have to apply and be sponsored by my company?
Once her visa is cancelled then so is yours. So your only option, if you want to stay in HK of course, is to get a working visa with your current company.
"Can I keep my working visa...?"
Yes, if in fact you have one, but it doesn't sound like you do. If you have a Dependant's Visa (which is what it sounds like you do) then as stated above that will expire when your wife's work visa is cancelled.