This is how I'd do it...

#!/usr/bin/perl

use autodie;  # Automatic errors on file problems.
use strict;

# This is the name of the file we want to modify.
my $filename = 'modify-file.txt';

# We're going to create a temporary file. This avoids us having
# to build up a potentially large string in memory.
my $tempname = $filename . '.tmp';

do
{
    # Open both files. Doing this using lexical file handles
    # within a "do" block means that when the end of the block
    # is reached, the files will be closed.
    open my $input_h, '<', $filename;   # input handle
    open my $output_h, '>', $tempname;  # output handle

    # Loop through each line of input.
    while (<$input_h>)
    {
        # Modify the line
        s/^>Sample_(\d+)_x(\d+)/>ID$1 $2/i;
        
        # Write it out.
        print $output_h $_;
    }
};

# Delete the original file.
unlink $filename while -f $filename;

# Rename the temporary file to the original filename.
rename $tempname => $filename;
[download]

This extraneous "do" is completely unnecessary. It actually "harms" by introducing an unnecessary level of indentation - which is a hindrance to readability.

It eliminates two calls to close and allows some lexical variables ($input_h and $output_h) to live in a smaller scope.

Indent it however you like; this ain't Python.

perl -E'sub Monkey::do{say$_,for@_,do{($monkey=[caller(0)]->[3])=~s{::}{ }and$monkey}}"Monkey say"->Monkey::do'

perl -i.bak -pe 's/^>Sample_([0-9]+)_x([0-9]+)$/ID$1 $2/' filename
[download]

Some issues with your effort

• the shebang is not an absolute path
• not using strict/warnings, Read this if you want to cut your development time in half!
• you're using <STDIN> instead of @ARGV (as in  perl myprogram.pl)
• you're reading from FILEHANDLE but your printing to STDOUT
• you're using FILEHANDLE instead of $filehandle
• your regular expression is case sensitive and it doesn't match your sample data

The general steps for editing a are

• read from original-file
• modify data
• write to new-file
• rename new-file to original-file

So you might write that as

#!/usr/bin/perl --
use strict; use warnings;
use autodie  2.1001;
use File::Temp qw/ tempfile /;
use File::Copy qw/ move /;
use autodie qw/ move /;
Main( @ARGV );
exit( 0 );
sub Main {
    return Usage() unless @_ ;
    for my $file ( @_) {
        print "Converting $file \n";
        ConvertFile( $file );
    }
}
sub ConvertFile {
    my $infilename = shift;
    my ($outfh, $outfilename) = tempfile();
    open my($infh), '<', $infilename; # autodie dies on error
    while( my $line = <$infh> ){
        chomp $line;
        $line =~ s/sample\_\d\_x?/ID\t/i;
        print $outfh $line, "\n";
    }
    close $infh;
    close $outfh;
    move( $outfilename, $infilename ); #  autodie dies on error
}
sub Usage {
    print <<"__USAGE__";

$0    
$0    modify/this/file 

perl ${\__FILE__}
perl ${\__FILE__}    modify/this/file 


__USAGE__
} ## end sub Usage

__END__
[download]

See use, autodie, open, File::Copy, File::Temp, strict, warnings, perlintro, perlretut, perlrequick, YAPE::Regex::Explain, Beginning Perl (free) Chapter 6: Files and Data, Modern Perl: Chapter 9: Managing Real Programs > Files

#!/usr/bin/perl -w
use strict;

my $ID = 1;
while (<DATA>)
{
   # this regex captures the trailing number if
   # the line starts with a ">"
   # the .*? means a "minimal match" of anything while 
   # allowing the rest of the regex to succeed.
   # the \n is counted as white space, a \s* character
   #
   if (my ($number) = $_ =~ /^>.*?(\d+)\s*$/)
   {
      print '>ID'.$ID++," $number\n"; 
   }
   else
   {
      print;
   }
}

=prints
>ID1 80
AGGGGGGGGGTTCCC
>ID2 85
TTTCCCGGGAAAA
>ID3 112
GGCCCCTTTGAGG
=cut

__DATA__
>Sample_1_x80
AGGGGGGGGGTTCCC
>Sample_2_x85
TTTCCCGGGAAAA
>sample_3_x112
GGCCCCTTTGAGG
[download]

while (<DATA>)
{
   if (my ($sample, $number) = $_ =~ /^>.*?(\d+).*?(\d+)\s*$/)
   {
      print '>ID'.$sample," $number\n"; 
   }
   else
   {
      print;
   }
}
[download]

Thanks everyone for your useful comments/codes!