From what I understand, jackknife and bootstrapping are frequentist methods for computing statistics (bias, variance, etc.) of an estimator.
Given a sample of my data and an estimator, and assuming little about the generative process, how can I compute the same statistics of my estimator using a Bayesian approach?
The bias and variance of an estimator are sampling properties of an estimator, so there is no difference between calculating the bias and variance of an estimator in sampling theory compared to Bayesian theory. That is, within both theories we look at the expected value and the variance of an estimator given the true parameter through the likelihood function: $\text{E}[ \text{Estimator} \mid \text{Truth}], \text{Var}[ \text{Estimator} \mid \text{Truth}]$.
With that said, Bayesians tend not to care too much about bias (they expect the prior to be reasonably close to the truth) and their estimates usually have lower variance because they are incorporating information from the the data and the prior (which is strictly speaking independent of the true parameters).