How to compare the acidity of this multicyclic hydrocarbon to the acidity of cyclopentadiene?
The method presented by Yusuf is a simplified model, which works best for well-behaved molecules. Unfortunately in this case it presents the 'correct' result based on incorrect assumptions. (Even though I did a calculation, I am not 100% convinced this is actually the correct result in the first place.)

In order for the resonance to stabilise a negative charge, the &pi;-systems must be aligned, which is the case for the deprotonation of cyclopentadiene, but it certainly is not true for the other molecule F (Loong said it's called 3a2H-benzo[3,4]pentaleno[2,1,6,5-jklm]fluorene). Here is the calculated structure at the ri-BP86/def2-SVP level of theory:



As you can see, it is domed, and there are three (almost) independent aromatic systems. Through deprotonation this will not change much. 

A reason for its relatively high acidity can be found in applying Coulson's Theorem, where the bond angle between two $\mathrm{sp}^x$ hybrid orbitals can be calculated from their angle $\omega_{ij}$ and their hybridisation indexes $\lambda_x$:
$$\cos(\omega_{ij}) = -\frac{1}{\sqrt{\lambda_i\lambda_j}}$$
Since the molecule is C3v symmetric, therefore $\lambda_i = \lambda_j \mathop{:=} x$, this simplifies to:
$$ x = -\sec(x)$$

The inner angle is $\approx\pu{102^\circ}$, therefore $x \approx 5$. The term for hybrid orbitals is actually just an abbreviation:
$$\mathrm{sp}^{x} = \mathrm{s}^{\frac{1}{x+1}}\mathrm{p}^{\frac{x}{x+1}}$$

Given that we just form linear combinations, we can see:
\begin{align}
 1\times\mathrm{s}, 3\times\mathrm{p} 
   &amp;\leadsto 3\times\mathrm{sp}^5, 1\times\mathrm{sp}
\end{align}
Given the rounding I have done, the remaining orbital is very roughly speaking an $\mathrm{sp}$ orbital; in the same ballpark as acetylene, from which we know that it has a very acidic proton.

The reason for the high acidity is therefore not the stabilisation of the conjugate base, but the destabilisation of the acid. This is the completely reverse reasoning as cor cyclopentadiene, in which the corresponding base is massively stabilised by resonance.



Given the above, how are you supposed to know, and how are you supposed to judge these compounds?

In my opinion, the exercise is something more on the level of quantum chemistry course and belongs into the canon of a masters student. While simply models will accidentally produce the correct result, it is important to see, that they completely fail for F. Structure-property relationships cannot be inferred from a 2D-graph of the molecule. To be honest, the reasoning above was also rather post-hoc, after I have calculated the molecule and analysed its structure.

I believe, (as long as you are not a human calculator), that there is no simple way to tell which of those molecules is more acidic; there is no easy way to weigh the stabilising effect of resonance against the destabilising effect of pyramidalization.

On the ri-BP86/def2-SVP level of theory (which is rather crude, but I don't have a supercomputer at home) it really comes out that F is more acidic. The following isodesmic reaction yields a difference in electronic energy of about $\Delta E_\mathrm{el} = \pu{-92.8 kJ mol-1}$.



Important: Even though the above sound reasonable, it still neglects plenty of things and has other shortcomings; to name the most important: thermal corrections, solvent effects, level of theory. 



Addendum:

Upon deprotonation the structure does not change very much (see larger image), it remains very domed, see the overlay of the structures below.



Judging from the three highest occupied molecular orbitals (HOMO, HOMO -1, HOMO -2) we can still presume some delocalisation of the negative charge into the adjacent aromatic rings.





I'm talking more about hybridisation, Bent's rule, and Coulson's theorem in the following: Are the lone pairs in water equivalent?