My question is simply: what happens if a character with a "web weapon" hits a monster with soar? Does the monster fall in the ground as caught in the web? Or maybe does it stay blocked in the air?

In our last session, Tobin used his falcon's claw and "caught" a razorwing, Well, we assumed that the first hypothesis was the right one: the razorwing fallen down to the ground 'cause its wings were blocked by the web. So, in the next turn, the razorwing was considered as adjacent (and not 4 squares away, like a soar monster should be) to his enemy (in this case, to Tobin himself).

Was this a correct way to solve that game issue?

No, it was not the correct way...

All Hail the Sometimes Infallable FAQ:

Q: What happens if a soaring creature is webbed? Does the soaring creature fall to the ground?
A: No. A soaring creature that is webbed may not spend movement points. Swooping and attacking do not cost movement points, so it may still do both freely.

ouch!

Well, thanks so much for the answer, but this rule has no sense at all, from my point of view...

TheHunterBoy said:

None of the rules for Soar makes sense, but yes the web ruling is even worse.

TheHunterBoy said:

There are quite a lot of rules in Descent that don't make sense. You'll get used to it. =P

Or you can change them together with your group to your liking. As long as you have fun