Depends on what you define as "survive".
There are buildings that survive centuries realively intact, even better if they are maintained over time. So there is a possibility of the walls and roofs "surviving".
As for the content, there are literally tens of thousands of clay tablets with cuneiform stored in hundreds of museums. A lot of them were once stored in the Library of Alexandria. Interestingly, the devastating fire that ultimately destroyed the library burned many of these tablets, hardening them and making them more likely to survive intact.
Most scripts were written on papyrus scrolls that would have decomposed under any less than perfect conditions. These conditions can only be achieved by completely sealing the papyri in an airtight container and never opening it again. Keeping content sealed is not exactly what you expect from a library, so it wouldn't have happened.
Since a papyrus scroll wouldn't stay intact forever, you would need to transcribe old scrolls onto new ones. With an estimated 40,000 to 400,000 scrolls stored in the library, you would need an army of scribes to constantly renew scrolls. Add in the fact that language and script change over time. There's the risk that a scribe simply couldn't read or understand a scroll written 300 years ago.
And then there's the content. Society, science and philosophy change over time. Transcribing a papyrus scroll is very time intense, so why would you transcribe a scroll that is hopelessly outdated if you could make your own name by writing your own book / scrolls?
All that is presuming the society at large is very stable and wealthy and in a time of peace.
Over such a long time, there are too many warlords wanting to burn down this symbol of his enemy just to make a statement. A state in financial need cannot employ hundreds of scribes to transfer squiggles and lines from one scroll to another. Natural catastrophes don't stop at the front door of a library.
To sum it up, the Royal Library of Alexandria could have survived as a building, but it's content wouldn't have.