Do not use an outdated OS, even with a modern browser.
Assuming that after that day I still use an updated browser, is it true that I'm still safe?
No, you cannot avoid browser-based security holes only by updating the browser. There are a few reasons for this. Primarily, the browser is not entirely self-contained. It makes use of operating system libraries, for example the system memory allocator. This allocator is designed to mitigate various memory corruption-related security issues. If the allocator is not kept up to date, memory exploitation bugs may be easier to perform against the browser, no matter how up to date the browser is.
Another reason is that browser security often relies on OS sandboxing features. A powerful browser exploit must be combined with a so-called sandbox escape. How easy that escape is depends on how secure the operating system is as well as how secure the browser is. By using an outdated operating system, your browser is being protected by out of date and potentially vulnerable security features.
Can it "patch" the OS-based security holes?
No. Patching operating system vulnerabilities requires elevated privileges, which a browser does not have. Even if it did, browsers are not designed to modify system settings or system files. There is no extension or web page you can go to that is able to patch security vulnerabilities in your OS.
Minor question: typically, how long would the browsers stop supporting abandoned OS?
Browser vendors typically publish when they will stop officially supporting a particular operating system. After that point, changes made to the browser that break on older systems will no longer be considered bugs and may not be fixed. Programs typically continue running on older systems for a very long time, however. They only stop working when they begin to rely on newer system APIs that aren't present in older versions. This is relatively rare. A browser should be able to run on an outdated operating system for many years, albeit not very securely, and without official support from the vendor. Most likely, as it begins to rely on newer and newer APIs, features in the browser will just start breaking one by one (especially security-related features) until it eventually does not start up at all.