Given a set of clients and a set of facilities with different priority levels in a metric space, the Budgeted Priorityp(1+ε)O(ndlogn)+(pε-1)pε-O(1)nO(1)n(3+ε)-Median problem aims to open a subset of facilities and connect each client to an opened facility with the same or a higher priority level, such that the number of opened facilities associated with each priority level is no more than a given upper limit, and the sum of the client-connection costs is minimized. In this paper, we present a data reduction-based approach for limiting the solution search space of the Budgeted Priorityp-Median problem, which yields a p(1+ε)O(ndlogn)+(pε-1)pε-O(1)nO(1)n(3+ε)-approximation algorithm running in p(1+ε)O(ndlogn)+(pε-1)pε-O(1)nO(1)n(3+ε) time in d-dimensional Euclidean space, where p(1+ε)O(ndlogn)+(pε-1)pε-O(1)nO(1)n(3+ε) is the size of the input instance and p is the maximal number of opened facilities. The previous best approximation ratio for this problem obtained in the same time is p(1+ε)O(ndlogn)+(pε-1)pε-O(1)nO(1)n(3+ε).