Degree of hydrolysis
I cannot understand how can degree of hydrolysis be very small
  compared to 1 when $\ce{NH4+}$ is a strong acid.


That is wrong.

You also wrote:


  It is a salt of a strong acid and a weak base.


Well, the salt of a strong acid and a weak base, i.e. the conjugated acid of that weak base is a weak acid.

In the case of the ammonium ion, $pK_a=9.25$, very weak indeed.

Also, the hydolysis reaction equilibrium is (because free $\ce{H+}$ don't exist in water and $\ce{NH4OH}$ is old notation, long obsolete):

$$\ce{NH4+ + H2O &lt;=&gt; NH3 + H3O+}$$

$h$ is negligibly small because this equilibrium leans very strongly to the left, so $[\ce{H3O+}]\approx 0$.