How to move string from one line to another

Using perl, if your file is not too big to be slurped in RAM memory:

perl -0 -e '                     # -0 slurp the whole file 
    $ = $, = "n";              # change record & list separator
    my @file = split /n/, <>;   # split the file on line per array value
    print @file[1,0,2..$#file]   # array slice
' file

If you want to replace in place, use

perl -i -e ......

$ printf '1m2n,pn' | ed -s file
line 2
line 1
line 3
line 4

The m command moves a line in the ed editor. 1m2 moves line 1 to line 2. ,p displays the modified buffer on standard output.

For an alternative approach which edits the file in-place:

ed -s file <<END_ED
1m2
w
q
END_ED

This performs the move, writes the result back to the file and quits.

sed -e '1~2{h;d};G'

That GNU sed expression in detail:

1~2 {  # lines 1,3,5,7 etc.
  h      # save line in the hold space
  d      # delete (don't print yet) and start next line
}
# only reached on lines 2,4,6,8 etc.
G        # retrieve from hold space, append to current line

I assume the backslash before the > is just a typo. You can use this bash script if your text file is well formed:

#!/bin/bash

while
    read -r a &&  # store one line to $a
    read -r   &&  # consume the blank line
    read -r b     # store another line to $b
do
    echo $b$a     # join those two lines
    read -r       # whatever, try to consume a blank line
done

Save it to s.sh.

You file content is like this:

A

A-

B

B-

C

C-

then run bash s.sh < file.txt you will get:

A-A
B-B
C-C