I'm just curious whether the expectation implied from a one-tailed test would somehow be considered a prior, and whether this is enough for it to be in the purview of Bayesian statistics.
No. The one-tailed t-test differs from the two-tailed t-test only in your choice of test statistic†—made so that it measures the discrepancy of the data with the null hypothesis in the way you're interested in. Otherwise it's the same, frequentist, method you follow—comparing the test statistic you actually observed with its distribution under a relevant sampling or random assignment scheme, assuming the truth of the null. Bayesian hypothesis testing requires you to assign prior, epistemic, probabilities to competing hypotheses, calculate the probability (density) of the data observed under a relevant sampling scheme for both hypotheses, & then obtain posterior probabilities for each hypothesis, conditional on the data observed, by applying Bayes' Theorem.
† That is $t$ (or $-t$) rather than $|t|$.