You can nearly do this with just <> and without touching @ARGV. First read eof then consider:

while( <> ) {
    print "First: $_";
    last if eof;    # Stop at end of /first/ file
}
while( <> ) {
    print;
}
[download]

But this actually does special processing of the first non-empty file, not the first file. Detecting that the first file is non-empty without directly modifying @ARGV is trickier:

my $args= @ARGV;
if(     ! eof()          # Prime the <> pump
    &&  @ARGV == $args-1 # Did we not move past the first file?
) {
    while( <> ) {
        print "First: $_";
        last if eof;
    }
}
while( <> ) {
    print;
}
[download]

In any case, I find it an interesting solution.

(Updated within seconds of posting to replace a near solution with a real solution.)