Prompted by the Python and Perl code, and just for the fun of it, here is a Haskell version (that probably shows that I am very much a beginner).

#!/usr/bin/runhaskell
main = interact $ unlines . map add20 . lines
add20 ('>':x) = ">" ++ x
add20 x       = reverse (take 20 (reverse x)) ++ x

But for such tasks one definitely should use shell commands as in the accepted answers.

Python:

#!/usr/bin/env python
import sys
with open(sys.argv[1], 'r') as f:
    for line in f:
        if line.startswith('>'):
            print (line.strip())
        else:
            print(line.strip()[-20:] + line.strip())

Save as append_last20.py, run as python append_last20.py input.fasta

Perl (Using my own input file similar to yours):

Input: seq1

>ID1:
ATCGATGCGCTATATCGCGATCGATGCTATGCgggggCTACGCTAGCTTCAG
>ID2:
AGCTAGCTGGATATGCTATCGATGCTAGCTACgggggCTAGCATGCTGATCG
>ID3:
GCGTAGCAGCATGCATGTATAGCTGCATCGAAgggggCATGCATGCTAGCTA

Code:

use strict;
use warnings;
my $regex = '.{20}$';
my $last20 = "";

my $filename = 'seq1';
open(my $fh, '<encoding(UTF-8)', $filename)
    or die "Could not open fie '$filename' $!";

while (my $row = <$fh>) {
    chomp $row;
    if($row =~ /^[a-zA-Z]/){
        if($row =~ /($regex)/){
            $last20 = $1;
            $last20 .= $row;
            print "$last20\n";
        }
    }
    else{
        print "$row\n";
    }
}

Output:

>ID1:
gggggCTACGCTAGCTTCAGATCGATGCGCTATATCGCGATCGATGCTATGCgggggCTACGCTAGCTTCAG
>ID2:
gggggCTAGCATGCTGATCGAGCTAGCTGGATATGCTATCGATGCTAGCTACgggggCTAGCATGCTGATCG
>ID3:
gggggCATGCATGCTAGCTAGCGTAGCAGCATGCATGTATAGCTGCATCGAAgggggCATGCATGCTAGCTA