In this paper, we investigate an intelligent reflecting surface (IRS) assisted full-duplex (FD) integrated sensing, communication and computing system. Specifically, an FD base station (BS) provides service for uplink and downlink transmission, and a local cache is connected to the BS through a backhaul link to store data. Meanwhile, active sensing elements are deployed on the IRS to receive target echo signals. On this basis, in order to evaluate the overall performance of the system under consideration, we propose a system utility maximization problem while ensuring the sensing quality, expressed as the difference between the sum of communication throughput, total computation bits (offloading bits and local computation bits) and the total backhaul cost for content delivery. This makes the problem difficult to solve due to the highly non-convex coupling of the optimization variables. To effectively solve this problem, we first design the most effective caching strategy. Then, we develop an algorithm based on weighted minimum mean square error, alternative direction method of multipliers, majorization-minimization framework, semi-definite relaxation techniques, and several complex transformations to jointly solve the optimization variables. Finally, simulation results are provided to verify the utility performance of the proposed algorithm and demonstrate the advantages of the proposed scheme compared with the baseline scheme.