This research investigates the properties of the posterior distribution of the gamma distribution, especially in the context of right-censored data. We establish necessary and sufficient conditions for determining when improper priors lead to proper posteriors. Additionally, we derive conditions to ascertain the finiteness of the posterior moments. The study addresses the challenges posed by censoring and delves into the application of various objective priors. We introduce a novel estimator for censored data, enhancing the efficiency of the Markov Chain Monte Carlo (MCMC) algorithm. Through a simulation study, we evaluate the performance of Bayesian estimators under different priors. Our methodology is applied to a dataset from the Cancer Genome Atlas, focussing on lung adenocarcinoma in patients over 70, offering valuable insights into disease progression and mortality patterns.