In networks with intermittent connectivity, such as mobile, aerial, and space systems, maintaining information freshness is complicated by time-varying arrivals, service disruptions, and interactions among traffic classes with different priorities. To capture these effects, we study a multi-priority single-server queue with time-varying arrivals and service rates under intermittent connectivity. Our main result shows that an appropriately selected collection of state-conditioned first moments closes exactly, leading to a finite-dimensional linear time-periodic Ordinary Differential Equation (ODE) system for the mean Age of Information (AoI) and mean Peak Age of Information (PAoI) of each priority class. For periodic arrival and service rates, we define a one-period state map by propagating the ODE over a single period, and use the periodicity condition to formulate the periodic steady state as a fixed point of this map. We then propose a fixed-point iteration algorithm and prove its convergence to the unique periodic steady state (PSS). Numerical results reveal that high-priority traffic can strongly reshape the service process seen by lower-priority classes.