One of the most important asymmetrical probability distributions that recently presented as an extension of the conventional exponential distribution is the alpha power exponential distribution. It may be compared to various asymmetrical well-known models, such as Weibull and gamma distributions. As a result, using an adaptive progressive Type-II hybrid censoring scheme, this paper investigates the estimation problems of the alpha power exponential distribution. Maximum likelihood and Bayesian methods are used to estimate unknown parameters, reliability, and hazard rate functions. Under the assumption of independent gamma priors and symmetric loss function, Bayesian estimators are examined. The Bayesian credible intervals and estimated confidence intervals of the relevant values are also calculated. The various estimating approaches are evaluated using a simulation study that considers various sample sizes and censoring schemes. Furthermore, numerous optimality criteria are examined, and the best progressive censoring schemes are offered. Finally, for an explanation, two real data sets from engineering and chemical fields are provided to show the applicability of the asymmetrical alpha power exponential distribution. The Bayesian method for estimating the parameters and reliability indices of the alpha power exponential distribution is recommended based on numerical results, especially when the number of observed data is small.