This work investigates the degrees of freedom (DoF) of a downlink cache-aided cellular network where the locations of base stations (BSs) are modeled as a grid topology and users within a grid cell can only communicate with four nearby BSs. We adopt a cache placement method with uncoded prefetching tailored for the network with partial connectivity. According to the overlapped degree of cached contents among BSs, we propose transmission schemes with no BS cooperation, partial BS cooperation, and full BS cooperation, respectively, for different cache sizes. In specific, the common cached contents among BSs are utilized to cancel some undesired signals by interference neutralization while interference alignment is used to coordinate signals of distinct cached contents. Our achievable results indicate that the reciprocal of per-user DoF of the cellular network decreases piecewise linearly with the normalized cache size $μ$ at each BS, and the gain of BS caching is more significant for the small cache region. Under the given cache placement scheme, we also provide an upper bound of per-user DoF and show that our achievable DoF is optimal when $μ\in\left[\frac{1}{2},1\right]$, and within an additive gap of $\frac{4}{39}$ to the optimum when $μ\in\left[\frac{1}{4},\frac{1}{2}\right)$.