One way would be to read the file into a string, put qq (with a delimiter that doesn't occur in the file) around it, and eval it

$APPNAME = "AnyName";

open my $fh, "<", "prog_file" or die $!;
my $lines; { local $/ = undef; $lines = <$fh>; }

print eval "qq|$lines|";
[download]

Another way would be to put $lines = qq| ... |; around the entire content of the file (in the file itself), and then use do

$APPNAME = "AnyName";

do "prog_file";
print $lines;
[download]

A third way would be to use regex substitution to replace strings like $APPNAME in the lines, but only if unescaped...

But maybe what you're really looking for is a templating system, like Text::Template, the Template Toolkit, etc...