In applications such as remote estimation and monitoring, update packets are transmitted by power-constrained devices using short-packet codes over wireless networks. Therefore, networks need to be end-to-end optimized using information freshness metrics such as age of information under transmit power and reliability constraints to ensure support for such applications. For short-packet coding, modelling and understanding the effect of block codeword length on transmit power and other performance metrics is important. To understand the above optimization for short-packet coding, we consider the optimal tradeoff problem between age of information and transmit power under reliability constraints for short packet point-to-point communication model with an exogenous packet generation process. In contrast to prior work, we consider scheduling policies that can possibly adapt the block-length or transmission time of short packet codes in order to achieve the optimal tradeoff. We characterize the tradeoff using a semi-Markov decision process formulation. We also obtain analytical upper bounds as well as numerical, analytical, and asymptotic lower bounds on the optimal tradeoff. We show that in certain regimes, such as high reliability and high packet generation rate, non-adaptive scheduling policies (fixed transmission time policies) are close-to-optimal. Furthermore, in a high-power or in a low-power regime, non-adaptive as well as state-independent randomized scheduling policies are order-optimal. These results are corroborated by numerical and simulation experiments. The tradeoff is then characterized for a wireless point-to-point channel with block fading as well as for other packet generation models (including an age-dependent packet generation model).