According to the documentation
System requirements for Office 2013[
^] then no, MS Access 2013 will
not work with Windows Server 2003 - it requires 2008 R2 or greater.
If you do upgrade the office version then your existing databases will
not be automatically changed, you will need to convert them yourself if this is required. Again, the documentation helps here ...
Convert a database to the .accdb file format - Access[
^]
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