Like the archaea?
Methane is abundant, let's say life forms when the planet is still mostly fire and volcanoes. High temperatures and no oxygen, perfect for archaebacterial life to be able to form.
Let's say then the planet cools down and eubacterial lifeforms start forming and they bring oxygen on the planet and new fauna.
Would archaebacterial life be able to survive and adapt to a cold planet full of oxygen?
Yes. The archaebacteria would persist in niches and ecosystems which did not have a lot of oxygen. They would be displaced by the new life forms in areas exposed to oxygen.
I could imagine similar with silicon and carbon based. The older life forms would be displaced from some places but persist in others where they were still better adapted than the newcomers. In some ecosystems there would be a mix of the two types.