---------(In Foo.pm)------------------
package Foo;
sub import {
    $Foo::var = shift;
}
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No, that's wrong. import() is called as a class method, thus the first argument will be the package name as a string, in this case: "Foo". You have to skip it, first, for example by using shift(), before you can get at the user supplied arguments.

One major difference between normal subs and class methods, apart from this extra first parameter (more trouble than much else, if you ask me), is that inheritance only works on methods, and not on plain subs. And the workings of Exporter are precisely based on this inheritance, where your package inherits the class method "import" from Exporter. Therefore, for Exporter to work, import() must be a method.