The regexp is correct. The problem is your string. Use single quote instead of double quote, to avoid interpolation. Try this code:

{
    my $name = "s@--__";
    print $name, "\n";
    if ($name =~ /\A[a-zA-Z][\w.-]*\z/) {
        print "match\n";
    } else {
        print "not match\n";
    }
}
{
    my $name = "s!!!@@@___";
    print $name, "\n";
    if ($name =~ /\A[a-zA-Z][\w.-]*\z/) {
        print "match\n";
    } else {
        print "not match\n";
    }
}
{
    my $name = 's@--__';
    print $name, "\n";
    if ($name =~ /\A[a-zA-Z][\w.-]*\z/) {
        print "match\n";
    } else {
        print "not match\n";
    }
}
{
    my $name = 's!!!@@@___';
    print $name, "\n";
    if ($name =~ /\A[a-zA-Z][\w.-]*\z/) {
        print "match\n";
    } else {
        print "not match\n";
    }
}
[download]

Which prints:

Possible unintended interpolation of @___ in string at math1.pl line 1
+1.
s-__
match
s!!!@@
not match
s@--__
not match
s!!!@@@___
not match
[download]