In my answer I'm going to try to keep the translation as closer to the original phrase as possible, because there're so many valid ways to say the very same. So I'm going to be concret.
In the translation there's a little mistake. The verb. It's past, but It's in the wrong form.
With each passing hour/day/year/etc ...
- Con cada día/hora/año que pasaba, sus conocimientos de español crecían.
The next one is another valid translation.
- Con cada día/hora/año que pasaba su dominio del español mejoraba.
A shorter way:
- Cada día/hora/año que pasaba su español mejoraba.
We can also change cada (each) by el paso de:
- Con el paso de los días/meses/años su español mejoraba.
Finally the shorter one:
Con los días/horas/años su español mejoraba.
We have elided that the time goes by, we just mentioned days/months/years for such meaning
Note that In spanish (at least in Spain), we speak about improving our language skills rather than increasing the knowledge about the language.
However, both ways (to improve and to increase) are interchangable in this phrase. Both are valid.
Note: passing is in continuous form. The -ing brings motion to the time. In spanish the motion has been introduced by que: que pasaba. This is the way we do reference explictly to the motion of the time.
If we want just to use the verb without que. We could also say:
- Pasaban los días/horas/años y su español mejoraba.
Semantically both say the very same. Sintactically both are differents. The last one would not be an acurated translation to the original one.