<IMG SRC="http://ilorentz.org/beenakker/MO/BesselK.png" WIDTH="400" />

The blue curve is the desired integral $\int_ 0^{a}\frac {e^{-x - \frac {1} {x}}} {x} dx$, the orange curve is the approximate answer $2K_0(2)a^3(1+a^3)^{-1}$. 

You could use this for a global approximation. In the small-$a$ region the asymptotics is $e^{-1/a}(a-2a^2)$, which is quite accurate up to about $a=0.2$, see plot below:

<IMG SRC="http://ilorentz.org/beenakker/MO/BesselK_2.png" WIDTH="400" />