How to simulate this randomized "finite" sum?
Let me give you some basic principles that seem helpful for your problem:


How to estimate an expected value. You want to estimate $\mathbb{E}[Y]$.  Here's a standard fact: if you can sample from the distribution of $Y$, then you can estimate $\mathbb{E}[Y]$.  Concisely, what you do is sample $n$ values according to the distribution of $Y$, then take their average; the result is a pretty good estimate for $\mathbb{E}[Y]$.  The larger the value of $n$, the more accurate your estimate will be.

With more mathematical precision: let $y_1,\dots,y_n$ be the result of sampling $n$ times from the distribution of $Y$.  This means that they come from i.i.d. r.v.'s with the same distribution as $Y$.  Now let $\hat{y} = (y_1+\dots + y_n)/n$ be the average of these values.  Then it follows that $\hat{y}$ is a good estimate for $\mathbb{E}[Y]$ (with high probability).  In particular, $\mathbb{E}[\hat{y}] = (\mathbb{E}[y_1]+\dots + \mathbb{E}[y_n])/n = (\mathbb{E}[Y]+\dots + \mathbb{E}[Y])/n = \mathbb{E}[Y]$, so $\hat{y}$ is an unbiased estimator for $\mathbb{E}[Y]$.  Also, by a similar calculation, $\text{Var}[\hat{y}] = \text{Var}[Y]/n$; the interpretation of this is that the error in your estimate of $\mathbb{E}[Y]$ will typically be roughly on the order of $\sigma/\sqrt{n}$, where $\sigma = \sqrt{\text{Var}[Y]}$ is the standard deviation of $Y$.  Thus, if $Y$ has variance that is not too large, then this approach gives a reasonably accurate estimate of $\mathbb{E}[Y]$.

In your case, this shows how to answer your second question: if you can sample from $Y$, this gives you one candidate way to estimate $\mathbb{E}[Y]$.  There are other ways to get an estimate, but this is a simple method that applies whenever you know how to sample from the underlying distribution.
How to sample from a sum of random length. The next question is, how to sample from $Y$?  The challenge you seem to be struggling with is how to deal with the fact that the definition of $Y$ is as a sum containing $N$ terms, but where here $N$ is a random variable.

It turns out there is a simple way to deal with this.  In particular, you can sample from $N$ by using the following process:


Sample from $N$, i.e., pick a random value for $N$ by sampling from a geometric distribution (since that's the distribution of $N$).
Now you have a fixed, constant value for $N$, so it becomes easy to sample from $Y$.  In particular, for this fixed value of $N$, sample a random value for $F_1,F_2,\dots,F_{N+1}$ (according to the distribution of these random variables).  Then, plug those values into the definition of $Y$; the result will be a random value for $Y$, picked according to the correct definition for $Y$.


In general, if you have a random variable that is defined by some stochastic process, then you can sample from it by simulating the process, sampling from random values whenever the process does.


So, let me spell out the implications for you.  This means that if you can figure out how to sample from the distribution of $F_i$, then you can use the above two techniques to sample from the distribution of $Y$ and thereby estimate the value of $\mathbb{E}[Y]$.  This reduces your problem to a smaller/simpler one: figuring out how to sample random values, distributed according to the distribution of $F_i$.