A new extended exponential distribution called the alpha-power exponential (APE) distribution is widely used for modeling various data compared to three well-known models, namely Weibull, gamma, and generalized-exponential distributions. This paper provides several approaches for estimating the APE parameters, reliability, and hazard rate functions using both classical and Bayesian methods when dealing with unified hybrid censored data. In the classical setup, we considered two methods called the maximum likelihood and maximum product of spacings. Moreover, using the delta method with observed Fisher information, asymptotic confidence intervals for the APE parameters or any function related to them are estimated. In the Bayes framework, using the squared-error loss with independent gamma density priors, we suggest two frequentist functions to develop the Bayes and highest posterior density credible interval estimators for all parameters. The Bayes estimation is performed by the Markov chain Monte Carlo sampling technique. Through extensive Monte Carlo simulations, based on various options of the effective sample sizes and threshold levels, the effectiveness of the proposed methodologies is compared. To show the applicability of the suggested estimators, two real-life datasets from the engineering domain are analyzed; the first consists of the yarn failure cycle and the other represents the failure times of the electronic components. The numerical analysis demonstrated that the APE model is suitable for analyzing the specified censored data. Furthermore, it is noted that the Bayesian method via the product of the spacing function yields more accurate results when a dataset under consideration is gathered from the proposed censoring.