To support the new and dominating traffic services - ultra-reliable and low latency communications (URLLC), the short-packet data communication techniques, such as finite blocklength coding (FBC), have been developed to guarantee the stringent delay and error-rate bounded quality-of-services (QoS) requirements for delay/age-sensitive wireless applications by using short-packet data communications. On the other hand, the age of information (AoI) has recently emerged as a new dimension of QoS performance metric in terms of the freshness of updated information for the delay/age-sensitive data transmissions. Since the status updates normally only consist of a small number of information bits and require ultra-low latency, integrating AoI with FBC creates another promising solution for supporting delay/age-sensitive URLLC services. However, how to characterize the relationships between AoI and delay over URLLC-enabled wireless networks has neither been well understood nor thoroughly studied in the finite blocklength regime. To overcome these challenges, we propose the joint optimization and tradeoff modeling for both peak AoI and delay-bound violation probabilities to support URLLC in the finite blocklength regime. First, we build up FBC based AoI-driven system models in the finite blocklength regime. Second, we apply the stochastic network calculus (SNC) to upper-bound both the peak AoI violation probability and delay-bound violation probability. Third, we jointly optimize the peak AoI violation probability and delay-bound violation probability and characterize their tradeoff in the finite blocklength regime. Finally, we conduct the extensive simulations to validate and evaluate our proposed AoI-driven schemes in the finite blocklength regime.