No. You don't "have" a server in a shared hosting environment, you share one with a whole lot of other people, and any versions of the major elements of the software stack (operating system, web server, PHP version, database server version, etc.) are the same for everybody. And since the only allowable hostname for the database server is "localhost", the option of having multiple, sepaarate database servers with different versions isn't there either.
If you need anything that is not part of the standard install (and can't run as a script under your user account), then you'll need to use a VPS (virtual private server), which will allow you, within limits, to install and run your own binaries.