番兵 '!' を特別扱いするのは型設計がよろしくないけど、手早く書くならこれでもいいか。
honeycomb.fs
let board =
  [| "!!!!!!!!!".ToCharArray();
     "!!!!TUVW!".ToCharArray();
     "!!!kHIJX!".ToCharArray();
     "!!jSBCKY!".ToCharArray();
     "!iRGADLZ!".ToCharArray();
     "!hQFEMa!!".ToCharArray();
     "!gPONb!!!".ToCharArray();
     "!fedc!!!!".ToCharArray();
     "!!!!!!!!!".ToCharArray() |]

let directions =
  [| (0,-1); (1,-1); (1,0); (0,1); (-1,1); (-1,0) |]

let folder (x,y,_) i =
  let (dx,dy) = directions.[i]
  let (x',y') = (x+dx,y+dy)
  let c' = board.[y'].[x']
  if c' = '!' then (x,y,c') else (x',y',c')

let solve (s:string) =
  let xs =
    s.ToCharArray()
    |> Array.map (fun c -> (int c)-(int '0'))
    |> Array.scan folder (4,4,board.[4].[4])
    |> Array.map (fun (_,_,c) -> c)
  new string(xs)

(* for test *)

type TestResult = Success | Failure

let test target expected =
  let actual = solve target
  printfn "%A" (if expected = actual then Success else Failure)

[<EntryPoint>]
let main args =
  test "135004" "ACDABHS"
  test "1" "AC"
  test "33333120" "AENc!!b!M"
  test "0" "AB"
  test "2" "AD"
  test "3" "AE"
  test "4" "AF"
  test "5" "AG"
  test "4532120" "AFQPOEMD"
  test "051455" "ABSHSj!"
  test "23334551" "ADMb!cdeO"
  test "22033251" "ADLKLa!ML"
  test "50511302122" "AGSjkTHTU!VW"
  test "000051" "ABHT!!!"
  test "1310105" "ACDKJW!V"
  test "50002103140" "AGSk!HU!IVIU"
  test "3112045" "AEDKYXKC"
  test "02021245535" "ABCIJW!JIHBS"
  test "014204" "ABIBCIB"
  test "255230" "ADAGAEA"
  test "443501" "AFPefgQ"
  test "022321" "ABCKLZ!"
  test "554452" "AGRh!!Q"
  test "051024" "ABSHTUH"
  test "524002" "AGAFGSB"
  test "54002441132" "AGQRjSRhRSGA"
  test "11010554312" "ACJV!!UTkSHI"
  test "23405300554" "ADMNEFOFGRi!"
  test "555353201" "AGRih!gPQG"
  test "22424105" "ADLMabaLD"
  test "11340202125" "ACJKDCKJX!!J"
  test "4524451" "AFQFPf!P"
  test "44434234050" "AFPf!!e!!Pgh"
  test "00554040132" "ABHk!j!i!jRG"
  test "3440403" "AEOePfgf"
  test "111130" "ACJW!XW"
  test "21133343125" "ADKXYZ!a!Z!L"
  test "353511" "AEFOPFA"
  test "22204115220" "ADLZYLY!KY!X"
  test "03013541" "ABABICBGB"
  test "101344" "ACIVJCA"
  test "2432541" "ADENbNdN"
  test "45332242015" "AFQPedc!!NME"
  test "215453" "ADKCAGF"
  test "45540523454" "AFQh!i!RQg!!"
  test "42434302545" "AFEOd!!ONOef"
  0