Sort:  
  1. Rxh5 Nf1
  2. Rh4 Ng3
  3. fxg6 fxg6
  4. c5 Kf7
  5. cxd6 cxd6
  6. Rc4 Re8
  7. Bf4 Nxe2
  8. Bxd6
    And white wins (even if only after 1. ... gxh5 there follows a spectacular checkmate).
  1. Rxh5 Nf1
  2. Rh4 Ng3
  3. fxg6 Nf5
  1. Rxh5 Nf1
  2. Rh4 Ng3
  3. fxg6 Nf5
  4. Rxh7 Rxh7
  5. gxh7+ Kxh7
  6. Bf4 (idea: b4-b5-b6)
    And white has a clear advantage.

Clear advantage isn't a win and it could take more than 13 moves.

It is a winning position, black will lose for sure.

The only critical move is 1. ... gxh5, because after every other move black simply has lost the knight, while his position is still bad. Therefore it is logical and most interesting to examine first what happens after 1. ... gxh5 which ends in a forced checkmate. Nobody wrote that white will always win within 13 moves.

You wrote the "Fastest solution". I can find a faster solution if I played both sides.

As all the discussion was about 1. ... gxh5 (and that rightly so), @iceblue found the fastest solution against that move. Of course white wins also against every other black move.

  1. Rxh5, Nf1
  2. Rh4, Ng3
  3. e4, gxf5
  4. Bh6, fxe4
  5. fxe4, Nf5
  6. Rg4+, Ng7
  7. Bxg7, black resigns

Or

  1. Rxh5, Nf1
  2. Rh4, Ng3
  3. e4, Nh5
  4. Rxh5!, gxh5
  5. Bh6, black resigns
  1. Rxh5 h6
  2. Rxh6 g5
  3. Rxh8 Kxh8
  4. Ke3

is it a clear win for black?

Its a loss for black. Bishop performs better than Knight in open games like this. Besides, White also has better position and material advantage.

Right, especially as the knight is trapped anyway in his variation ... :)