Fluid antenna systems (FAS) have emerged as a revolutionary technology offering enhanced spatial diversity within a compact form factor. Concurrently, unmanned aerial vehicles (UAVs) are integral to future networks, necessitating channel models that capture both multipath fading and shadowing. This letter presents a novel performance analysis of a UAV-to-ground link, where the receiver is equipped with an $N$-port FAS operating over the challenging double-shadowing fading channel. By adapting a tractable eigenvalue-based approximation for the correlated FAS ports, we derive new analytical expressions for the end-to-end signal-to-noise ratio statistics, namely the cumulative distribution function and the probability density function. Based on these statistics, we present exact integral expressions for the outage probability, average bit error rate, and average channel capacity. We further derive new, tractable closed-form solutions for the average bit error rate and capacity for the practical dual-rank, independent but non-identically distributed case. Finally, a key asymptotic analysis reveals that the system achieves a multiplicative diversity order of $G_d = M \times d$, which is precisely the product of the FAS spatial rank $M$ and the intrinsic channel diversity order $d$. Simulation results are provided to validate the high accuracy of our entire theoretical framework.