AAAA
BBBB
CCCC
DDDD

1234
5678
9012
3456

EEEE

7890

AAAA 1234
BBBB 5678
CCCC 9012
DDDD 3456

EEEE 7890

use warnings;
use strict;

## Check arguments.
die qq[Usage: perl $0 <input-file>n] unless @ARGV == 1;

my (@alpha, @digit);

while ( <> ) {
        ## Omit blank lines.
        next if m/As*Z/;

        ## Remove leading and trailing spaces.
        s/As*//;
        s/s*Z//;

        ## Save alphanumeric fields and fields with
        ## only digits to different arrays.
        if ( m/A[[:alpha:]]+Z/ ) {
                push @alpha, $_;
        }
        elsif ( m/A[[:digit:]]+Z/ ) {
                push @digit, $_;
        }
}

## Get same positions from both arrays and print them
## in the same line.
for my $i ( 0 .. $#alpha ) {
        printf qq[%s %sn], $alpha[ $i ], $digit[ $i ];
}

AAAA
BBBB
CCCC
DDDD

1234
5678
9012
3456

EEEE

7890

perl script.pl infile

AAAA 1234
BBBB 5678
CCCC 9012
DDDD 3456
EEEE 7890

awk -v RS="" '{for(i=1; i<=NF; i++) a[i]=$i
  getline
  for(i=1; i<=NF; i++) print a[i] " " $i
  print ""}' file

<input sed -nr '/^[A-Z]{4}$/,/^$/w out1
                /^[0-9]{4}$/,/^$/w out2'
paste -d' ' out1 out2 |sed 's/^ $//'

paste -d' ' <(sed -nr '/^[A-Z]{4}$/,/^$/p' input)
            <(sed -nr '/^[0-9]{4}$/,/^$/p' input) | sed 's/^ $//'

perl -M5.010 -lne '
  given ($_) {
    when (/^[[:alpha:]]+$/) {push @alpha, $_}
    when (/^d+$/) {say shift(@alpha), " ", $_}
    default {say}
  }
'

$ pr -mt <(grep -i "^[a-z]" file.txt) <(grep -i "^[0-9]" file.txt)
AAAA                    1234
BBBB                    5678
CCCC                    9012
DDDD                    3456
EEEE                    7890